Capacity of Clustered Distributed Storage

Jy-yong Sohn,  Beongjun Choi,  Sung Whan Yoon,  and Jaekyun Moon The authors are with the School of Electrical Engineering, Korea Advanced Institute of Science and Technology, Daejeon, 34141, Republic of Korea (e-mail: {jysohn1108, bbzang10, shyoon8}@kaist.ac.kr, jmoon@kaist.edu). A part of this paper was presented [1] at the IEEE Conference on Communications (ICC), Paris, France, May 21-25, 2017. This work is in part supported by the National Research Foundation of Korea under Grant No. 2016R1A2B4011298, and in part supported by the ICT R&D program of MSIP/IITP [2016-0-00563, Research on Adaptive Machine Learning Technology Development for Intelligent Autonomous Digital Companion].
Abstract

A new system model reflecting the clustered structure of distributed storage is suggested to investigate interplay between storage overhead and repair bandwidth as storage node failures occur. Large data centers with multiple racks/disks or local networks of storage devices (e.g. sensor network) are good applications of the suggested clustered model. In realistic scenarios involving clustered storage structures, repairing storage nodes using intact nodes residing in other clusters is more bandwidth-consuming than restoring nodes based on information from intra-cluster nodes. Therefore, it is important to differentiate between intra-cluster repair bandwidth and cross-cluster repair bandwidth in modeling distributed storage. Capacity of the suggested model is obtained as a function of fundamental resources of distributed storage systems, namely, node storage capacity, intra-cluster repair bandwidth and cross-cluster repair bandwidth. The capacity is shown to be asymptotically equivalent to a monotonic decreasing function of number of clusters, as the number of storage nodes increases without bound. Based on the capacity expression, feasible sets of required resources which enable reliable storage are obtained in a closed-form solution. Specifically, it is shown that the cross-cluster traffic can be minimized to zero (i.e., intra-cluster local repair becomes possible) by allowing extra resources on storage capacity and intra-cluster repair bandwidth, according to the law specified in the closed-form. The network coding schemes with zero cross-cluster traffic are defined as intra-cluster repairable codes, which are shown to be a class of the previously developed locally repairable codes.

Index Terms:
Capacity, Distributed storage, Network coding

I Introduction

Many enterprises, including Google, Facebook, Amazon and Microsoft, use cloud storage systems in order to support massive amounts of data storage requests from clients. In the emerging Internet-of-Thing (IoT) era, the number of devices which generate data and connect to the network increases exponentially, so that efficient management of data center becomes a formidable challenge. However, since cloud storage systems are composed of inexpensive commodity disks, failure events occur frequently, degrading the system reliability [2].

In order to ensure reliability of cloud storage, distributed storage systems (DSSs) with erasure coding have been considered to improve tolerance against storage node failures [3, 4, 5, 6, 7, 8]. In such systems, the original file is encoded and distributed into multiple storage nodes. When a node fails, a newcomer node regenerates the failed node by contacting a number of survived nodes. This causes traffic burden across the network, taking up significant repair bandwidth. Earlier distributed storage systems utilized the 3-replication code: the original file was replicated three times, and the replicas were stored in three distinct nodes. The 3-replication coded systems require the minimum repair bandwidth, but incur high storage overhead. Reed-Solomon (RS) codes are also used (e.g. HDFS-RAID in Facebook [9]), which allow minimum storage overhead; however, RS-coded systems suffer from high repair bandwidth.

The pioneering work of [10] on distributed storage systems focused on the relationship between two required resources, the storage capacity α𝛼\alpha of each node and the repair bandwidth γ𝛾\gamma, when the system aims to reliably store a file \mathcal{M} under node failure events. The optimal (α,γ)𝛼𝛾(\alpha,\gamma) pairs are shown to have a fundamental trade-off relationship, to satisfy the maximum-distance-separable (MDS) property (i.e., any k𝑘k out of n𝑛n storage nodes can be accessed to recover the original file) of the system. Moreover, the authors of [10] obtained capacity 𝒞𝒞\mathcal{C}, the maximum amount of reliably storable data, as a function of α𝛼\alpha and γ𝛾\gamma. The authors related the failure-repair process of a DSS with the multi-casting problem in network information theory, and exploited the fact that a cut-set bound is achievable by network coding [11]. Since the theoretical results of [10], explicit network coding schemes [12, 13, 14] which achieve the optimal (α,γ)𝛼𝛾(\alpha,\gamma) pairs have also been suggested. These results are based on the assumption of homogeneous systems, i.e., each node has the same storage capacity and repair bandwidth.

However, in real data centers, storage nodes are dispersed into multiple clusters (in the form of disks or racks) [15, 7, 8], allowing high reliability against both node and rack failure events. In this clustered system, repairing a failed node gives rise to both intra-cluster and cross-cluster repair traffic. While the current data centers have abundant intra-rack communication bandwidth, cross-rack communication is typically limited. According to [16], nearly a 180180180TB of cross-rack repair bandwidth is required everyday in the Facebook warehouse, limiting cross-rack communication for foreground map-reduce jobs. Moreover, surveys [17, 18, 19] on network traffic within data centers show that cross-rack communication is oversubscribed; the available cross-rack communication bandwidth is typically 5205205-20 times lower than the intra-rack bandwidth in practical systems. Thus, a new system model which reflects the imbalance between intra- and cross-cluster repair bandwidths is required.

I-A Main Contributions

This paper suggests a new system model for clustered DSS to reflect the clustered nature of real distributed storage systems wherein an imbalance exists between intra- and cross-cluster repair burdens. This model can be applied to not only large data centers, but also local networks of storage devices such as the sensor networks or home clouds which are expected to be prevalent in the IoT era. This model is also more general in the sense that when the intra- and cross-cluster repair bandwidths are set to be equal, the resulting structure reduces to the original DSS model of [10]. This paper only considers recovering a single node failure at a time, as in [10]. The main contributions of this paper can be seen as twofold: one is the derivation of a closed-form expression for capacity, and the other is the analysis on feasible sets of system resources which enable reliable storage.

I-A1 Closed-form Expression for Capacity

Under the setting of functional repair, storage capacity 𝒞𝒞\mathcal{C} of the clustered DSS is obtained as a function of node storage capacity α𝛼\alpha, intra-cluster repair bandwidth βIsubscript𝛽𝐼\beta_{I} and cross-cluster repair bandwidth βcsubscript𝛽𝑐\beta_{c}. The existence of the cluster structure manifested as the imbalance between intra/cross-cluster traffics makes the capacity analysis challenging; Dimakis’ proof in [10] cannot be directly extended to handle the problem at hand. We show that symmetric repair (obtaining the same amount of information from each helper node) is optimal in the sense of maximizing capacity given the storage node size and total repair bandwidth, as also shown in [20] for the case of varying repair bandwidth across the nodes. However, we stress that in most practical scenarios, the need is greater for reducing cross-cluster communication burden, and we show that this is possible by trading with reduced overall storage capacity and/or increasing intra-repair bandwidth. Based on the derived capacity expression, we analyzed how the storage capacity 𝒞𝒞\mathcal{C} changes as a function of L𝐿L, the number of clusters. It is shown that the capacity is asymptotically equivalent to C¯¯𝐶\underline{C}, some monotonic decreasing function of L𝐿L.

I-A2 Analysis on Feasible (α,βI,βc)𝛼subscript𝛽𝐼subscript𝛽𝑐(\alpha,\beta_{I},\beta_{c}) Points

Given the need for reliably storing file \mathcal{M}, the set of required resource pairs, node storage capacity α𝛼\alpha, intra-cluster repair bandwidth βIsubscript𝛽𝐼\beta_{I} and cross-cluster repair bandwidth βcsubscript𝛽𝑐\beta_{c}, which enables 𝒞(α,βI,βc)𝒞𝛼subscript𝛽𝐼subscript𝛽𝑐\mathcal{C}(\alpha,\beta_{I},\beta_{c})\geq\mathcal{M} is obtained in a closed-form solution. In the analysis, we introduce ϵ=βc/βIitalic-ϵsubscript𝛽𝑐subscript𝛽𝐼\epsilon=\beta_{c}/\beta_{I}, a useful parameter which measures the ratio of the cross-cluster repair burden (per node) to the intra-cluster burden. This parameter represents how scarce the available cross-cluster bandwidth is, compared to the abundant intra-cluster bandwidth. We here stress that the special case of ϵ=0italic-ϵ0\epsilon=0 corresponds to the scenario where repair is done only locally via intra-cluster communication, i.e., when a node fails the repair process requires intra-cluster traffic only without any cross-cluster traffic. Thus, the analysis on the ϵ=0italic-ϵ0\epsilon=0 case provides a guidance on the network coding for data centers for the scenarios where the available cross-cluster (cross-rack) bandwidth is very scarce.

Similar to the non-clustered case of [10], the required node storage capacity and the required repair bandwidth show a trade-off relationship. In the trade-off curve, two extremal points - the minimum-bandwidth-regenerating (MBR) point and the minimum-storage-regenerating (MSR) point - have been further analyzed for various ϵitalic-ϵ\epsilon values. Moreover, from the analysis on the trade-off curve, it is shown that the minimum storage overhead α=/k𝛼𝑘\alpha=\mathcal{M}/k is achievable if and only if ϵ1nkitalic-ϵ1𝑛𝑘\epsilon\geq\frac{1}{n-k}. This implies that in order to reliably store file \mathcal{M} with minimum storage α=/k𝛼𝑘\alpha=\mathcal{M}/k, sufficiently large cross-cluster repair bandwidth satisfying ϵ1nkitalic-ϵ1𝑛𝑘\epsilon\geq\frac{1}{n-k} is required. Finally, for the scenarios with the abundant intra-cluster repair bandwidth, the minimum required cross-cluster repair bandwidth βcsubscript𝛽𝑐\beta_{c} to reliably store file \mathcal{M} is obtained as a function of node storage capacity α𝛼\alpha.

I-B Related Works

Several researchers analyzed practical distributed storage systems with a goal in mind to reflect the non-homogeneous nature of storage nodes [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31]. A heterogeneous model was considered in [20, 21] where the storage capacity and the repair bandwidth for newcomer nodes are generally non-uniform. Upper/lower capacity bounds for the heterogeneous DSS are obtained in [20]. An asymmetric repair process is considered in [22], coining the terms, cheap and expensive nodes, based on the amount of data that can be transfered to any newcomer. The authors of [23] considered a flexible distributed storage system where the amount of information from helper nodes may be non-uniform, as long as the total repair bandwidth is bounded from above. The view points taken in these works are different from ours in that we adopt a notion of cluster and introduce imbalance between intra- and cross-cluster repair burdens.

Recently, some researchers considered the clustered structure of data centers [1, 24, 25, 26, 27, 28, 29, 30, 31]. Some recent works [24, 25, 26, 27] provided new system models for clustered DSS and shed light on fundamental aspects of the suggested system. In [24], the idea of [22] is developed to a two-rack system, by setting the communication burden within a rack much lower than the burden across different racks, similar to our analysis. However, the authors of [24] only considered systems with two racks, while the current paper considers a general setting of L𝐿L racks (clusters), and provides mathematical analysis on how the number of clusters (i.e., the dispersion of nodes) affects the capacity of clustered distributed storage. Similar to the present paper, the authors of [25, 26] obtained the capacity of clustered distributed storage, and provided capacity-achieving regenerating coding schemes. However, the coding schemes considered in [25, 26] do not satisfy the MDS property, and the capacity expression is obtained for limited scenarios when intra-cluster repair bandwidth βIsubscript𝛽𝐼\beta_{I} is set to its maximum value. In contrast, the current paper provides the capacity expression for general values of βI,βcsubscript𝛽𝐼subscript𝛽𝑐\beta_{I},\beta_{c} parameters, and analyzes the behavior of capacity as a function of the ratio ϵ=βc/βIitalic-ϵsubscript𝛽𝑐subscript𝛽𝐼\epsilon=\beta_{c}/\beta_{I} between intra- and cross-cluster repair bandwidths. Moreover, unlike in the previous work, the capacity-achieving coding schemes (whose existence is shown) here satisfy the MDS property. In [27], the security issue in clustered distributed storage systems is considered, and the maximum amount of securely storable data in the existence of passive eavesdroppers is obtained.

There also have been some recent works [28, 29, 30, 31] on network code design appropriate for clustered distributed storage. Motivated by the limited available cross-rack repair bandwidth in real data centers, the work of [28] provides a network coding scheme which minimizes the cross-rack bandwidth in clustered distributed storage systems. However, the suggested coding scheme is applicable for some limited (n,k,L)𝑛𝑘𝐿(n,k,L) parameters and the minimum storage overhead (α=/k𝛼𝑘\alpha=\mathcal{M}/k) setting. On the other hand, the current paper provides the capacity expression for general (n,k,L,α)𝑛𝑘𝐿𝛼(n,k,L,\alpha) setting, and proves the existence of capacity-achieving coding scheme. The authors of [29] proposed coding schemes tolerant to rack failure events in multi-rack storage systems, but have not addressed the imbalance between intra- and cross-cluster repair burdens in node failure events, which is an important aspect of the current paper. The authors of [30] considers the scenario of having grouped (clustered) storage nodes where nodes in the same group are more accessible to each other, compared to the nodes in other groups. However, the focus is different to the present paper: [30] focuses on the code construction which has the minimum amount of accessed data (called the optimal access property), while the scope of the present paper is on finding the optimal trade-off between the node storage capacity and the repair bandwidth, as a function of the ratio ϵitalic-ϵ\epsilon of intra- and cross-cluster communication burdens. Finally, a locally repairable code which can repair arbitrary node within each group is suggested in [31]; this code can suppress the inter-group repair bandwidth to zero. However, the coding scheme is suggested for the ϵ=0italic-ϵ0\epsilon=0 case only, while the present paper provides the capacity expression for general 0ϵ10italic-ϵ10\leq\epsilon\leq 1, and proves the existence of an optimal coding scheme.

Compared to the conference version [1] of the current work, this paper provides the formal proofs for the capacity expression, and obtains the feasible (α,γ)𝛼𝛾(\alpha,\gamma) region for 0ϵ10italic-ϵ10\leq\epsilon\leq 1 setting111The reason why the present paper considers this regime is provided in Section III-B. (only ϵ=0italic-ϵ0\epsilon=0 is considered in [1]). The present paper also shows the behavior of capacity as a function of L𝐿L, the number of clusters, and provides the sufficient and necessary conditions on ϵ=βc/βIitalic-ϵsubscript𝛽𝑐subscript𝛽𝐼\epsilon=\beta_{c}/\beta_{I}, to achieve the minimum storage overhead α=/k𝛼𝑘\alpha=\mathcal{M}/k. Finally, the asymptotic behaviors of the MBR/MSR points are investigated in this paper, and the connection between what we call the intra-cluster repairable codes and the existing locally repairable codes [32] is revealed.

I-C Organization

This paper is organized as follows. Section II reviews preliminary materials about distributed storage systems and the information flow graph, an efficient tool for analyzing DSS. Section III proposes a new system model for the clustered DSS, and derives a closed-form expression for the storage capacity of the clustered DSS. The behavior of the capacity curves is also analyzed in this section. Based on the capacity expression, Section IV provides results on the feasible resource pairs which enable reliable storage of a given file. Further research topics on clustered DSS are discussed in Section V, and Section VI draws conclusions.

II Background

II-A Distributed Storage System

Distributed storage systems can maintain reliability by means of erasure coding [33]. The original data file is spread into n𝑛n potentially unreliable nodes, each with storage size α𝛼\alpha. When a node fails, it is regenerated by contacting d<n𝑑𝑛d<n helper nodes and obtaining a particular amount of data, β𝛽\beta, from each helper node. The amount of communication burden imposed by one failure event is called the repair bandwidth, denoted as γ=dβ𝛾𝑑𝛽\gamma=d\beta. When the client requests a retrieval of the original file, assuming all failed nodes have been repaired, access to any k<n𝑘𝑛k<n out of n𝑛n nodes must guarantee a file recovery. The ability to recover the original data using any k<n𝑘𝑛k<n out of n𝑛n nodes is called the maximal-distance-separable (MDS) property. Distributed storage systems can be used in many applications such as large data centers, peer-to-peer storage systems and wireless sensor networks [10].

II-B Information Flow Graph

Information flow graph is a useful tool to analyze the amount of information flow from source to data collector in a DSS, as utilized in [10]. It is a directed graph consisting of three types of nodes: data source SS\mathrm{S}, data collector DCDC\mathrm{DC}, and storage nodes xisuperscript𝑥𝑖x^{i} as shown in Fig. 1. Storage node xisuperscript𝑥𝑖x^{i} can be viewed as consisting of input-node xinisuperscriptsubscript𝑥𝑖𝑛𝑖x_{in}^{i} and output-node xoutisuperscriptsubscript𝑥𝑜𝑢𝑡𝑖x_{out}^{i}, which are responsible for the incoming and outgoing edges, respectively. xinisuperscriptsubscript𝑥𝑖𝑛𝑖x_{in}^{i} and xoutisuperscriptsubscript𝑥𝑜𝑢𝑡𝑖x_{out}^{i} are connected by a directed edge with capacity identical to the storage size α𝛼\alpha of node xisuperscript𝑥𝑖x^{i}.

Data from source S𝑆S is stored into n𝑛n nodes. This process is represented by n𝑛n edges going from SS\mathrm{S} to {xi}i=1nsuperscriptsubscriptsuperscript𝑥𝑖𝑖1𝑛\{x^{i}\}_{i=1}^{n}, where each edge capacity is set to infinity. A failure/repair process in a DSS can be described as follows. When a node xjsuperscript𝑥𝑗x^{j} fails, a new node xn+1superscript𝑥𝑛1x^{n+1} joins the graph by connecting edges from d𝑑d survived nodes, where each edge has capacity β𝛽\beta. After all repairs are done, data collector DCDC\mathrm{DC} chooses arbitrary k𝑘k nodes to retrieve data, as illustrated by the edges connected from k𝑘k survived nodes with infinite edge capacity. Fig. 1 gives an example of information flow graph representing a distributed storage system with n=4,k=3,d=3formulae-sequence𝑛4formulae-sequence𝑘3𝑑3n=4,k=3,d=3.

Refer to caption
Figure 1: Information flow graph (n=4,k=3,d=3formulae-sequence𝑛4formulae-sequence𝑘3𝑑3n=4,k=3,d=3)
Refer to caption
Figure 2: Clustered distributed storage system (n=15,L=3formulae-sequence𝑛15𝐿3n=15,L=3)

II-C Notation used in the paper

This paper requires many notations related to graphs, because it deals with information flow graphs. Here we provide the definition of each notation used in the paper. For the given system parameters, we denote 𝒢𝒢\mathcal{G} as the set of all possible information flow graphs. A graph G𝒢𝐺𝒢G\in\mathcal{G} is denoted as G=(V,E)𝐺𝑉𝐸G=(V,E) where V𝑉V is the set of vertices and E𝐸E is the set of edges in the graph. For a given graph G𝒢𝐺𝒢G\in\mathcal{G}, we call a set cE𝑐𝐸c\subset E of edges as cut-set [34] if it satisfies the following: every directed path from SS\mathrm{S} to DCDC\mathrm{DC} includes at least one edge in c𝑐c. An arbitrary cut-set c𝑐c is usually denoted as c=(U,U¯)𝑐𝑈¯𝑈c=(U,\mkern 1.5mu\overline{\mkern-1.5muU\mkern-1.5mu}\mkern 1.5mu) where UV𝑈𝑉U\subset V and U¯=VU¯𝑈𝑉𝑈\mkern 1.5mu\overline{\mkern-1.5muU\mkern-1.5mu}\mkern 1.5mu=V\setminus U (the complement of U𝑈U) satisfy the following: the set of edges from U𝑈U to U¯¯𝑈\mkern 1.5mu\overline{\mkern-1.5muU\mkern-1.5mu}\mkern 1.5mu is the given cut-set c𝑐c. The set of all cut-sets available in G𝐺G is denoted as C(G)𝐶𝐺C(G). For a graph G𝒢𝐺𝒢G\in\mathcal{G} and a cut-set cC(G)𝑐𝐶𝐺c\in C(G), we denote the sum of edge capacities for edges in c𝑐c as w(G,c)𝑤𝐺𝑐w(G,c), which is called the cut-value of c𝑐c.

A vector is denoted as v using the bold notation. For a vector v, the transpose of the vector is denoted as vTsuperscriptv𝑇\textbf{v}^{T}. A set is denoted as X={x1,x2,,xk}𝑋subscript𝑥1subscript𝑥2subscript𝑥𝑘X=\{x_{1},x_{2},\cdots,x_{k}\}, while a sequence x1,x2,,xNsubscript𝑥1subscript𝑥2subscript𝑥𝑁x_{1},x_{2},\cdots,x_{N} is denoted as (xn)n=1Nsuperscriptsubscriptsubscript𝑥𝑛𝑛1𝑁(x_{n})_{n=1}^{N}, or simply (xn)subscript𝑥𝑛(x_{n}). For given sequences (an)subscript𝑎𝑛(a_{n}) and (bn)subscript𝑏𝑛(b_{n}), we use the term “ansubscript𝑎𝑛a_{n} is asymptotically equivalent to bnsubscript𝑏𝑛b_{n}[35] if and only if

limnanbn=1.subscript𝑛subscript𝑎𝑛subscript𝑏𝑛1\lim\limits_{n\rightarrow\infty}\frac{a_{n}}{b_{n}}=1. (1)

We utilize a useful notation:

𝟙i=j={1, if i=j0, otherwise.subscript1𝑖𝑗cases1 if 𝑖𝑗0 otherwise.\mathds{1}_{i=j}=\begin{cases}1,&\text{ if }i=j\\ 0,&\text{ otherwise.}\end{cases}

For a positive integer n𝑛n, we use [n]delimited-[]𝑛[n] as a simplified notation for the set {1,2,,n}12𝑛\{1,2,\cdots,n\}. For a non-positive integer n𝑛n, we define [n]=delimited-[]𝑛[n]=\emptyset. Each storage node is represented as either xt=(xint,xoutt)superscript𝑥𝑡superscriptsubscript𝑥𝑖𝑛𝑡superscriptsubscript𝑥𝑜𝑢𝑡𝑡x^{t}=(x_{in}^{t},x_{out}^{t}) as defined in Section II-B, or N(i,j)𝑁𝑖𝑗N(i,j) as defined in (A.2). Finally, important parameters used in this paper are summarized in Table I.

TABLE I: Parameters used in this paper
n𝑛n number of storage nodes
k𝑘k number of DC-contacting nodes
L𝐿L number of clusters
nI=n/Lsubscript𝑛𝐼𝑛𝐿n_{I}=n/L number of nodes in a cluster
dI=nI1subscript𝑑𝐼subscript𝑛𝐼1d_{I}=n_{I}-1 number of intra-cluster helper nodes
dc=nnIsubscript𝑑𝑐𝑛subscript𝑛𝐼d_{c}=n-n_{I} number of cross-cluster helper nodes
𝔽qsubscript𝔽𝑞\mathbb{F}_{q} base field which contains each symbol
α𝛼\alpha storage capacity of each node
βIsubscript𝛽𝐼\beta_{I} intra-cluster repair bandwidth (per node)
βcsubscript𝛽𝑐\beta_{c} cross-cluster repair bandwidth (per node)
γI=dIβIsubscript𝛾𝐼subscript𝑑𝐼subscript𝛽𝐼\gamma_{I}=d_{I}\beta_{I} intra-cluster repair bandwidth
γc=dcβcsubscript𝛾𝑐subscript𝑑𝑐subscript𝛽𝑐\gamma_{c}=d_{c}\beta_{c} cross-cluster repair bandwidth
γ=γI+γc𝛾subscript𝛾𝐼subscript𝛾𝑐\gamma=\gamma_{I}+\gamma_{c} repair bandwidth
ϵ=βc/βIitalic-ϵsubscript𝛽𝑐subscript𝛽𝐼\epsilon=\beta_{c}/\beta_{I} ratio of βcsubscript𝛽𝑐\beta_{c} to βIsubscript𝛽𝐼\beta_{I} (0ϵ10italic-ϵ10\leq\epsilon\leq 1)
ξ=γc/γ𝜉subscript𝛾𝑐𝛾\xi=\gamma_{c}/\gamma ratio of γcsubscript𝛾𝑐\gamma_{c} to γ𝛾\gamma (0ξ<10𝜉10\leq\xi<1)
R=k/n𝑅𝑘𝑛R=k/n ratio of k𝑘k to n𝑛n (0<R10𝑅10<R\leq 1)

III Capacity of Clustered DSS

III-A Clustered Distributed Storage System

A distributed storage system with multiple clusters is shown in Fig. 2. Data from source SS\mathrm{S} is stored at n𝑛n nodes which are grouped into L𝐿L clusters. The number of nodes in each cluster is fixed and denoted as nI=n/Lsubscript𝑛𝐼𝑛𝐿n_{I}=n/L. The storage size of each node is denoted as α𝛼\alpha. When a node fails, a newcomer node is regenerated by contacting dIsubscript𝑑𝐼d_{I} helper nodes within the same cluster, and dcsubscript𝑑𝑐d_{c} helper nodes from other clusters. This paper considers functional repair[33] in the regeneration process; the newcomer node may store different content from that of the failed node, while maintaining the MDS property of the code. The amount of data a newcomer node receives within the same cluster is γI=dIβIsubscript𝛾𝐼subscript𝑑𝐼subscript𝛽𝐼\gamma_{I}=d_{I}\beta_{I} (each node equally contributes to βIsubscript𝛽𝐼\beta_{I}), and that from other clusters is γc=dcβcsubscript𝛾𝑐subscript𝑑𝑐subscript𝛽𝑐\gamma_{c}=d_{c}\beta_{c} (each node equally contributes to βcsubscript𝛽𝑐\beta_{c}). Fig. 3 illustrates an example of information flow graph representing the repair process in a clustered DSS.

Refer to caption
Figure 3: Repair process in clustered DSS (n=4,L=2,dI=1,dc=2formulae-sequence𝑛4formulae-sequence𝐿2formulae-sequencesubscript𝑑𝐼1subscript𝑑𝑐2n=4,L=2,d_{I}=1,d_{c}=2)

III-B Assumptions for the System

We assume that dcsubscript𝑑𝑐d_{c} and dIsubscript𝑑𝐼d_{I} have the maximum possible values (dc=nnI,dI=nI1formulae-sequencesubscript𝑑𝑐𝑛subscript𝑛𝐼subscript𝑑𝐼subscript𝑛𝐼1d_{c}=n-n_{I},d_{I}=n_{I}-1), since this is the capacity-maximizing choice, as formally stated in the following proposition. The proof of the proposition is in Appendix G-A.

Proposition 1.

Consider a clustered distributed storage system with given γ𝛾\gamma and γcsubscript𝛾𝑐\gamma_{c}. Then, setting both dIsubscript𝑑𝐼d_{I} and dcsubscript𝑑𝑐d_{c} to their maximum values maximizes storage capacity.

Note that the authors of [10] already showed that in the non-clustered scenario with given repair bandwidth γ𝛾\gamma, maximizing the number of helper nodes d𝑑d is the capacity-maximizing choice. Here, we are saying that a similar property also holds for clustered scenario considered in the present paper. Under the setting of the maximum number of helper nodes, the overall repair bandwidth for a failure event is denoted as

γ=γI+γc=(nI1)βI+(nnI)βc𝛾subscript𝛾𝐼subscript𝛾𝑐subscript𝑛𝐼1subscript𝛽𝐼𝑛subscript𝑛𝐼subscript𝛽𝑐\gamma=\gamma_{I}+\gamma_{c}=(n_{I}-1)\beta_{I}+(n-n_{I})\beta_{c} (2)

Data collector DCDC\mathrm{DC} contacts any k𝑘k out of n𝑛n nodes in the clustered DSS. Given that the typical intra-cluster communication bandwidth is larger than the cross-cluster bandwidth in real systems, we assume

βIβcsubscript𝛽𝐼subscript𝛽𝑐\beta_{I}\geq\beta_{c}

throughout the present paper; this assumption limits our interest to ϵ1italic-ϵ1\epsilon\leq 1. Moreover, motivated by the security issue, we assume that a file cannot be retrieved entirely by contacting any single cluster having nIsubscript𝑛𝐼n_{I} nodes. Thus, the number k𝑘k of nodes contacted by the data collector satisfies

k𝑘\displaystyle k >nI.absentsubscript𝑛𝐼\displaystyle>n_{I}. (3)

We also assume

L2𝐿2L\geq 2 (4)

which holds for most real DSSs. Usually, all storage nodes cannot be squeezed in a single cluster, i.e., L=1𝐿1L=1 rarely happens in practical systems, to prevent losing everything when the cluster is destroyed. Note that many storage systems [7, 8, 16] including those of Facebook uses L=n𝐿𝑛L=n, i.e., every storage node reside in different racks (clusters), to tolerate the rack failure events. Finally, according to [16], nearly 98%percent9898\% of data recoveries in real systems deal with single node recovery. In other words, the portion of simultaneous multiple nodes failure events is small. Therefore, the present paper focuses single node failure events.

III-C The closed-form solution for Capacity

Consider a clustered DSS with fixed n,k,L𝑛𝑘𝐿n,k,L values. In this model, we want to find the set of feasible parameters (α,βI,βc𝛼subscript𝛽𝐼subscript𝛽𝑐\alpha,\beta_{I},\beta_{c}) which enables storing data of size \mathcal{M}. In order to find the feasible set, min-cut analysis on the information flow graph is required, similar to [10]. Depending on the failure-repair process and k𝑘k nodes contacted by DCDC\mathrm{DC}, various information flow graphs can be obtained.

Let 𝒢𝒢\mathcal{G} be the set of all possible flow graphs. Consider a graph G𝒢superscript𝐺𝒢G^{*}\in\mathcal{G} with minimum min-cut, the construction of which is specified in Appendix A. Based on the max-flow min-cut theorem in [11], the maximum information flow from source to data collector for arbitrary G𝒢𝐺𝒢G\in\mathcal{G} is greater than or equal to

𝒞(α,βI,βc)min-cut of G,𝒞𝛼subscript𝛽𝐼subscript𝛽𝑐min-cut of superscript𝐺\mathcal{C}(\alpha,\beta_{I},\beta_{c})\coloneqq\text{min-cut of }G^{*},

which is called the capacity of the system. In order to send data \mathcal{M} from the source to the data collector, 𝒞𝒞\mathcal{C}\geq\mathcal{M} should be satisfied. Moreover, if 𝒞𝒞\mathcal{C}\geq\mathcal{M} is satisfied, there exists a linear network coding scheme [11] to store a file with size \mathcal{M}. Therefore, the set of (α,βI,βc)𝛼subscript𝛽𝐼subscript𝛽𝑐(\alpha,\beta_{I},\beta_{c}) points which satisfies 𝒞𝒞\mathcal{C}\geq\mathcal{M} is feasible in the sense of reliably storing the original file of size \mathcal{M}. Now, we state our main result in the form of a theorem which offers a closed-form solution for the capacity 𝒞(α,βI,βc)𝒞𝛼subscript𝛽𝐼subscript𝛽𝑐\mathcal{C}(\alpha,\beta_{I},\beta_{c}) of the clustered DSS. Note that setting βI=βcsubscript𝛽𝐼subscript𝛽𝑐\beta_{I}=\beta_{c} reduces to capacity of the non-clustered DSS obtained in [10].

Theorem 1.

The capacity of the clustered distributed storage system with parameters (n,k,L,α,βI,βc)𝑛𝑘𝐿𝛼subscript𝛽𝐼subscript𝛽𝑐(n,k,L,\alpha,\beta_{I},\beta_{c}) is

𝒞(α,βI,βc)=i=1nIj=1gimin{α,ρiβI+(nρijm=1i1gm)βc},𝒞𝛼subscript𝛽𝐼subscript𝛽𝑐superscriptsubscript𝑖1subscript𝑛𝐼superscriptsubscript𝑗1subscript𝑔𝑖𝛼subscript𝜌𝑖subscript𝛽𝐼𝑛subscript𝜌𝑖𝑗superscriptsubscript𝑚1𝑖1subscript𝑔𝑚subscript𝛽𝑐\mathcal{C}(\alpha,\beta_{I},\beta_{c})=\sum_{i=1}^{n_{I}}\sum_{j=1}^{g_{i}}\min\{\alpha,\rho_{i}\beta_{I}+(n-\rho_{i}-j-\sum_{m=1}^{i-1}g_{m})\beta_{c}\}, (5)

where

ρisubscript𝜌𝑖\displaystyle\rho_{i} =nIi,absentsubscript𝑛𝐼𝑖\displaystyle=n_{I}-i, (6)
gmsubscript𝑔𝑚\displaystyle g_{m} ={knI+1,m(kmodnI)knI,otherwise.absentcases𝑘subscript𝑛𝐼1𝑚𝑘modsubscript𝑛𝐼𝑘subscript𝑛𝐼𝑜𝑡𝑒𝑟𝑤𝑖𝑠𝑒\displaystyle=\begin{cases}\lfloor\frac{k}{n_{I}}\rfloor+1,&m\leq(k\ \mathrm{mod}\ n_{I})\\ \lfloor\frac{k}{n_{I}}\rfloor,&otherwise.\end{cases} (7)

The proof is in Appendix A. Note that the parameters used in the statement of Theorem 1 have the following property, the proof of which is in Appendix G-B.

Proposition 2.

For every (i,j)𝑖𝑗(i,j) with i[nI],j[gi]formulae-sequence𝑖delimited-[]subscript𝑛𝐼𝑗delimited-[]subscript𝑔𝑖i\in[n_{I}],j\in[g_{i}], we have

ρiβI+(nρijm=1i1gm)βcγ.subscript𝜌𝑖subscript𝛽𝐼𝑛subscript𝜌𝑖𝑗superscriptsubscript𝑚1𝑖1subscript𝑔𝑚subscript𝛽𝑐𝛾\rho_{i}\beta_{I}+(n-\rho_{i}-j-\sum_{m=1}^{i-1}g_{m})\beta_{c}\leq\gamma. (8)

Moreover,

m=1nIgm=ksuperscriptsubscript𝑚1subscript𝑛𝐼subscript𝑔𝑚𝑘\sum_{m=1}^{n_{I}}g_{m}=k (9)

holds.

III-D Relationship between 𝒞𝒞\mathcal{C} and ϵ=βc/βIitalic-ϵsubscript𝛽𝑐subscript𝛽𝐼\epsilon=\beta_{c}/\beta_{I}

In this subsection, we analyze the capacity of a clustered DSS as a function of an important parameter

ϵβc/βI,italic-ϵsubscript𝛽𝑐subscript𝛽𝐼\epsilon\coloneqq\beta_{c}/\beta_{I}, (10)

the cross-cluster repair burden per intra-cluster repair burden. In Fig. 4, capacity is plotted as a function of ϵitalic-ϵ\epsilon. From (2), the total repair bandwidth can be expressed as

γ𝛾\displaystyle\gamma =γI+γc=(nI1)βI+(nnI)βcabsentsubscript𝛾𝐼subscript𝛾𝑐subscript𝑛𝐼1subscript𝛽𝐼𝑛subscript𝑛𝐼subscript𝛽𝑐\displaystyle=\gamma_{I}+\gamma_{c}=(n_{I}-1)\beta_{I}+(n-n_{I})\beta_{c}
=(nI1+(nnI)ϵ)βI.absentsubscript𝑛𝐼1𝑛subscript𝑛𝐼italic-ϵsubscript𝛽𝐼\displaystyle=\Big{(}n_{I}-1+(n-n_{I})\epsilon\Big{)}\beta_{I}. (11)

Using this expression, the capacity is expressed as

𝒞(ϵ)=i=1nIj=1gimin{α,(nρijm=1i1gm)ϵ+ρi(nnI)ϵ+nI1γ}.𝒞italic-ϵsuperscriptsubscript𝑖1subscript𝑛𝐼superscriptsubscript𝑗1subscript𝑔𝑖𝛼𝑛subscript𝜌𝑖𝑗superscriptsubscript𝑚1𝑖1subscript𝑔𝑚italic-ϵsubscript𝜌𝑖𝑛subscript𝑛𝐼italic-ϵsubscript𝑛𝐼1𝛾\mathcal{C}(\epsilon)=\sum_{i=1}^{n_{I}}\sum_{j=1}^{g_{i}}\min\Big{\{}\alpha,\frac{(n-\rho_{i}-j-\sum_{m=1}^{i-1}g_{m})\epsilon+\rho_{i}}{(n-n_{I})\epsilon+n_{I}-1}\gamma\Big{\}}. (12)

For fair comparison on various ϵitalic-ϵ\epsilon values, capacity is calculated for a fixed (n,k,L,α,γ𝑛𝑘𝐿𝛼𝛾n,k,L,\alpha,\gamma) set. The capacity is an increasing function of ϵitalic-ϵ\epsilon as shown in Fig. 4. This implies that for given resources α𝛼\alpha and γ𝛾\gamma, allowing a larger βcsubscript𝛽𝑐\beta_{c} (until it reaches βIsubscript𝛽𝐼\beta_{I}) is always beneficial, in terms of storing a larger file. For example, under the setting in Fig. 4, allowing βc=βIsubscript𝛽𝑐subscript𝛽𝐼\beta_{c}=\beta_{I} (i.e., ϵ=1italic-ϵ1\epsilon=1) can store =4848\mathcal{M}=48, while setting βc=0subscript𝛽𝑐0\beta_{c}=0 (i.e., ϵ=0italic-ϵ0\epsilon=0) cannot achieve the same level of storage. This result is consistent with the previous work on asymmetric repair in [20], which proved that the symmetric repair maximizes capacity. Therefore, when the total communication amount γ𝛾\gamma is fixed, a loss of storage capacity is the cost we need to pay in order to reduce the communication burden βcsubscript𝛽𝑐\beta_{c} across different clusters.

Refer to caption
Figure 4: Capacity as a function of ϵitalic-ϵ\epsilon, under the setting of n=100,k=85,L=10,α=1,γ=1formulae-sequence𝑛100formulae-sequence𝑘85formulae-sequence𝐿10formulae-sequence𝛼1𝛾1n=100,k=85,L=10,\alpha=1,\gamma=1

III-E Relationship between 𝒞𝒞\mathcal{C} and L𝐿L

In this subsection, we analyze the capacity of a clustered DSS as a function of L𝐿L, the number of clusters. For fair comparison, (n,k,α,γ𝑛𝑘𝛼𝛾n,k,\alpha,\gamma) values are fixed for calculating capacity. In Fig. 5, capacity curves for two scenarios are plotted over a range of L𝐿L values. First, the solid line corresponds to the scenario when the system has abundant cross-rack bandwidth resources γcsubscript𝛾𝑐\gamma_{c}. In this ideal scenario which does not suffer from the over-subscription problem, the system can store =8080\mathcal{M}=80 irrespective of the dispersion of nodes.

However, consider a practical situation where the available cross-rack bandwidth is scarce compared to the intra-rack bandwidth; for example, ξ=γc/γ=1/5𝜉subscript𝛾𝑐𝛾15\xi=\gamma_{c}/\gamma=1/5. The dashed line in Fig. 5 corresponds to this scenario where the system has not enough cross-rack bandwidth resources. In this practical scenario, reducing L𝐿L (i.e., gathering the storage nodes into a smaller number of clusters) increases capacity. However, note that sufficient dispersion of data into a fair number of clusters is typically desired, in order to guarantee the reliability of storage in rack-failure events. Finding the optimal number Lsuperscript𝐿L^{*} of clusters in this trade-off relationship remains as an important topic for future research.

Refer to caption
Figure 5: Capacity as a function of L𝐿L, under the setting of n=100,k=80,α=1,γ=10formulae-sequence𝑛100formulae-sequence𝑘80formulae-sequence𝛼1𝛾10n=100,k=80,\alpha=1,\gamma=10
Refer to caption
Figure 6: An example of DSS with dispersion ratio σ=5/3𝜎53\sigma=5/3, when the parameters are set to L=3,nI=5,n=nIL=15formulae-sequence𝐿3formulae-sequencesubscript𝑛𝐼5𝑛subscript𝑛𝐼𝐿15L=3,n_{I}=5,n=n_{I}L=15

In Fig. 5, the capacity is a monotonic decreasing function of L𝐿L when the system suffers from an over-subscription problem. However, in general (n,k,α,γ)𝑛𝑘𝛼𝛾(n,k,\alpha,\gamma) parameter settings, capacity is not always a monotonic decreasing function of L𝐿L. Theorem 2 illustrates the behavior of capacity as L𝐿L varies, focusing on the special case of γ=α.𝛾𝛼\gamma=\alpha. Before formally stating the next main result, we need to define

σLnI=L2n,𝜎𝐿subscript𝑛𝐼superscript𝐿2𝑛\sigma\coloneqq\frac{L}{n_{I}}=\frac{L^{2}}{n}, (13)

the dispersion factor of a clustered storage system, as illustrated in Fig. 6. In the two-dimensional representation of a clustered distributed storage system, L𝐿L represents the number of rows (clusters), while nI=n/Lsubscript𝑛𝐼𝑛𝐿n_{I}=n/L represents the number of columns (nodes in each cluster). The dispersion factor σ𝜎\sigma is the ratio of the number of rows to the number of columns. If L𝐿L increases for a fixed nIsubscript𝑛𝐼n_{I}, then σ𝜎\sigma grows and the nodes become more dispersed into multiple clusters.

Now we state our second main result, which is about the behavior of 𝒞𝒞\mathcal{C} versus L𝐿L.

Theorem 2.

Consider the γ=α𝛾𝛼\gamma=\alpha case when σ,γ,R𝜎𝛾𝑅\sigma,\gamma,R and ξ𝜉\xi are fixed. In the asymptotic regime of large n𝑛n, capacity 𝒞(α,βI,βc)𝒞𝛼subscript𝛽𝐼subscript𝛽𝑐\mathcal{C}(\alpha,\beta_{I},\beta_{c}) is asymptotically equivalent to

C¯=k2(γ+nkn(11/L)γc),¯𝐶𝑘2𝛾𝑛𝑘𝑛11𝐿subscript𝛾𝑐\underline{C}=\frac{k}{2}\left(\gamma+\frac{n-k}{n(1-1/L)}\gamma_{c}\right), (14)

a monotonically decreasing function of L𝐿L. This can also be stated as

𝒞C¯similar-to𝒞¯𝐶\mathcal{C}\sim\underline{C} (15)

as n𝑛n\rightarrow\infty for a fixed σ𝜎\sigma.

Note that under the setting of Theorem 2, we have

nIsubscript𝑛𝐼\displaystyle n_{I} =nL=nnσ=Θ(n),absent𝑛𝐿𝑛𝑛𝜎Θ𝑛\displaystyle=\frac{n}{L}=\frac{n}{\sqrt{n\sigma}}=\Theta(\sqrt{n}),
k𝑘\displaystyle k =nR=Θ(n),α=γ=constant,formulae-sequenceabsent𝑛𝑅Θ𝑛𝛼𝛾constant\displaystyle=nR=\Theta(n),\quad\alpha=\gamma=\text{constant},
βIsubscript𝛽𝐼\displaystyle\beta_{I} =γInI1=Θ(1n),βc=γcnnI=Θ(1n)formulae-sequenceabsentsubscript𝛾𝐼subscript𝑛𝐼1Θ1𝑛subscript𝛽𝑐subscript𝛾𝑐𝑛subscript𝑛𝐼Θ1𝑛\displaystyle=\frac{\gamma_{I}}{n_{I}-1}=\Theta(\frac{1}{\sqrt{n}}),\quad\beta_{c}=\frac{\gamma_{c}}{n-n_{I}}=\Theta(\frac{1}{n})

in the asymptotic regime of large n𝑛n. The proof of Theorem 2 is based on the following two lemmas.

Lemma 1.

In the case of γ=α𝛾𝛼\gamma=\alpha, capacity 𝒞(α,βI,βc)𝒞𝛼subscript𝛽𝐼subscript𝛽𝑐\mathcal{C}(\alpha,\beta_{I},\beta_{c}) is upper/lower bounded as

C¯𝒞(α,βI,βc)C¯+δ¯𝐶𝒞𝛼subscript𝛽𝐼subscript𝛽𝑐¯𝐶𝛿\underline{C}\leq\mathcal{C}(\alpha,\beta_{I},\beta_{c})\leq\underline{C}+\delta (16)

where

δ𝛿\displaystyle\delta =nI2(βIβc)/8absentsuperscriptsubscript𝑛𝐼2subscript𝛽𝐼subscript𝛽𝑐8\displaystyle=n_{I}^{2}(\beta_{I}-\beta_{c})/8 (17)

and C¯¯𝐶\underline{C} is defined in (14).

Lemma 2.

In the case of γ=α𝛾𝛼\gamma=\alpha, C¯¯𝐶\underline{C} and δ𝛿\delta defined in (14), (17) satisfy

C¯¯𝐶\displaystyle\underline{C} =Θ(n),absentΘ𝑛\displaystyle=\Theta(n),
δ𝛿\displaystyle\delta =O(nI)absent𝑂subscript𝑛𝐼\displaystyle=O(n_{I})

when γ,R𝛾𝑅\gamma,R and ξ𝜉\xi are fixed.

The proofs of these lemmas are in Appendix H. Here we provide the proof of Theorem 2 by using Lemmas 1 and 2.

proof (of Theorem 2).

From Lemma 2,

δ/C¯=O(1/L)=O(1/nσ)0𝛿¯𝐶𝑂1𝐿𝑂1𝑛𝜎0\delta/\underline{C}=O(1/L)=O(\sqrt{1/n\sigma})\rightarrow 0 (18)

as n𝑛n\rightarrow\infty for a fixed σ𝜎\sigma. Moreover, dividing (16) by C¯¯𝐶\underline{C} results in

1𝒞/C¯1+δ/C¯.1𝒞¯𝐶1𝛿¯𝐶1\leq\mathcal{C}/\underline{C}\leq 1+\delta/\underline{C}. (19)

Putting (18) into (19) completes the proof. ∎

IV Discussion on Feasible (α,βI,βc𝛼subscript𝛽𝐼subscript𝛽𝑐\alpha,\beta_{I},\beta_{c})

In the previous section, we obtained the capacity of the clustered DSS. This section analyzes the feasible (α,βI,βc𝛼subscript𝛽𝐼subscript𝛽𝑐\alpha,\beta_{I},\beta_{c}) points which satisfy 𝒞(α,βI,βc)𝒞𝛼subscript𝛽𝐼subscript𝛽𝑐\mathcal{C}(\alpha,\beta_{I},\beta_{c})\geq\mathcal{M} for a given file size \mathcal{M}. Using

γ=γI+γc=(nI1)βI+(nnI)βc𝛾subscript𝛾𝐼subscript𝛾𝑐subscript𝑛𝐼1subscript𝛽𝐼𝑛subscript𝑛𝐼subscript𝛽𝑐\gamma=\gamma_{I}+\gamma_{c}=(n_{I}-1)\beta_{I}+(n-n_{I})\beta_{c}

in (2), the behavior of the feasible set of (α,γ)𝛼𝛾(\alpha,\gamma) points can be observed. Essentially, the feasible points demonstrate a trade-off relationship. Two extreme points – minimum storage regenerating (MSR) point and minimum bandwidth regenerating (MBR) point – of the trade-off has been analyzed. Moreover, a special family of network codes which satisfy ϵ=0italic-ϵ0\epsilon=0, which we call the intra-cluster repairable codes, is compared with the locally repairable codes considered in [32]. Finally, the set of feasible (α,βc)𝛼subscript𝛽𝑐(\alpha,\beta_{c}) points is discussed, when the system allows maximum βIsubscript𝛽𝐼\beta_{I}.

IV-A Set of Feasible (α,γ)𝛼𝛾(\alpha,\gamma) Points

We provide a closed-form solution for the set of feasible (α,γ)𝛼𝛾(\alpha,\gamma) points which enable reliable storage of data \mathcal{M}. Based on the range of ϵitalic-ϵ\epsilon defined in (10), the set of feasible points show different behaviors as stated in Corollary 1.

Corollary 1.

Consider a clustered DSS for storing data \mathcal{M}, when ϵ=βc/βIitalic-ϵsubscript𝛽𝑐subscript𝛽𝐼\epsilon=\beta_{c}/\beta_{I} satisfies 0ϵ10italic-ϵ10\leq\epsilon\leq 1. For any γγ(α)𝛾superscript𝛾𝛼\gamma\geq\gamma^{*}(\alpha), the data \mathcal{M} can be reliably stored, i.e., 𝒞𝒞\mathcal{C}\geq\mathcal{M}, while it is impossible to reliably store data \mathcal{M} when γ<γ(α)𝛾superscript𝛾𝛼\gamma<\gamma^{*}(\alpha). The threshold function γ(α)superscript𝛾𝛼\gamma^{*}(\alpha) can be obtained as:

  1. 1.

    if 1nkϵ11𝑛𝑘italic-ϵ1\frac{1}{n-k}\leq\epsilon\leq 1

    γ(α)={,α(0,Mk)Mtαst,α[Mt+1+st+1yt+1,Mt+styt),(t=k1,k2,,1)Ms0,α[Ms0,)superscript𝛾𝛼cases𝛼0𝑀𝑘𝑀𝑡𝛼subscript𝑠𝑡𝛼𝑀𝑡1subscript𝑠𝑡1subscript𝑦𝑡1𝑀𝑡subscript𝑠𝑡subscript𝑦𝑡otherwise𝑡𝑘1𝑘21𝑀subscript𝑠0𝛼𝑀subscript𝑠0\gamma^{*}(\alpha)=\begin{cases}\infty,&\ \alpha\in(0,\frac{M}{k})\\ \frac{M-t\alpha}{s_{t}},&\ \alpha\in[\frac{M}{t+1+s_{t+1}y_{t+1}},\frac{M}{t+s_{t}y_{t}}),\\ &\ \ \ \ \ \ \ \ \ \ (t=k-1,k-2,\cdots,1)\\ \frac{M}{s_{0}},&\ \alpha\in[\frac{M}{s_{0}},\infty)\end{cases} (20)
  2. 2.

    Otherwise (if 0ϵ<1nk0italic-ϵ1𝑛𝑘0\leq\epsilon<\frac{1}{n-k})

    γ(α)={,α(0,Mτ+i=τ+1kzi)ταsτ,α[Mτ+i=τ+1kzi,τ+sτyτ)Mtαst,α[Mt+1+st+1yt+1,Mt+styt),(t=τ1,τ2,,1)Ms0,α[Ms0,)superscript𝛾𝛼cases𝛼0𝑀𝜏superscriptsubscript𝑖𝜏1𝑘subscript𝑧𝑖𝜏𝛼subscript𝑠𝜏𝛼𝑀𝜏superscriptsubscript𝑖𝜏1𝑘subscript𝑧𝑖𝜏subscript𝑠𝜏subscript𝑦𝜏𝑀𝑡𝛼subscript𝑠𝑡𝛼𝑀𝑡1subscript𝑠𝑡1subscript𝑦𝑡1𝑀𝑡subscript𝑠𝑡subscript𝑦𝑡otherwise𝑡𝜏1𝜏21𝑀subscript𝑠0𝛼𝑀subscript𝑠0\gamma^{*}(\alpha)=\begin{cases}\infty,&\ \alpha\in(0,\frac{M}{\tau+\sum_{i=\tau+1}^{k}z_{i}})\\ \frac{\mathcal{M}-\tau\alpha}{s_{\tau}},&\ \alpha\in[\frac{M}{\tau+\sum_{i=\tau+1}^{k}z_{i}},\frac{\mathcal{M}}{\tau+s_{\tau}y_{\tau}})\\ \frac{M-t\alpha}{s_{t}},&\ \alpha\in[\frac{M}{t+1+s_{t+1}y_{t+1}},\frac{M}{t+s_{t}y_{t}}),\\ &\ \ \ \ \ \ \ \ \ \ (t=\tau-1,\tau-2,\cdots,1)\\ \frac{M}{s_{0}},&\ \alpha\in[\frac{M}{s_{0}},\infty)\end{cases} (21)

where

τ𝜏\displaystyle\tau =max{t{0,1,,k1}:zt1}absent:𝑡01𝑘1subscript𝑧𝑡1\displaystyle=\max\{t\in\{0,1,\cdots,k-1\}:z_{t}\geq 1\} (22)
stsubscript𝑠𝑡\displaystyle s_{t} ={i=t+1kzi(nI1)+ϵ(nnI),t=0,1,,k10,t=kabsentcasessuperscriptsubscript𝑖𝑡1𝑘subscript𝑧𝑖subscript𝑛𝐼1italic-ϵ𝑛subscript𝑛𝐼𝑡01𝑘10𝑡𝑘\displaystyle=\begin{cases}\frac{\sum_{i=t+1}^{k}z_{i}}{(n_{I}-1)+\epsilon(n-n_{I})},&t=0,1,\cdots,k-1\\ 0,&t=k\end{cases} (23)
ytsubscript𝑦𝑡\displaystyle y_{t} =(nI1)+ϵ(nnI)zt,absentsubscript𝑛𝐼1italic-ϵ𝑛subscript𝑛𝐼subscript𝑧𝑡\displaystyle=\frac{(n_{I}-1)+\epsilon(n-n_{I})}{z_{t}}, (24)
ztsubscript𝑧𝑡\displaystyle z_{t} ={(nnIt+ht)ϵ+(nIht),t[k],t=0absentcases𝑛subscript𝑛𝐼𝑡subscript𝑡italic-ϵsubscript𝑛𝐼subscript𝑡𝑡delimited-[]𝑘𝑡0\displaystyle=\begin{cases}(n-n_{I}-t+h_{t})\epsilon+(n_{I}-h_{t}),&t\in[k]\\ \infty,&t=0\end{cases} (25)
htsubscript𝑡\displaystyle h_{t} =min{s[nI]:l=1sglt},absent:𝑠delimited-[]subscript𝑛𝐼superscriptsubscript𝑙1𝑠subscript𝑔𝑙𝑡\displaystyle=\min\{s\in[n_{I}]:\sum_{l=1}^{s}g_{l}\geq t\}, (26)

and {gl}l=1nIsuperscriptsubscriptsubscript𝑔𝑙𝑙1subscript𝑛𝐼\{g_{l}\}_{l=1}^{n_{I}} is defined in (7).

Refer to caption
Figure 7: Optimal tradeoff between node storage size α𝛼\alpha and total repair bandwidth γ𝛾\gamma, under the setting of n=15,k=8,L=3,=8formulae-sequence𝑛15formulae-sequence𝑘8formulae-sequence𝐿38n=15,k=8,L=3,\mathcal{M}=8
Refer to caption
Figure 8: Optimal tradeoff between node storage size α𝛼\alpha and cross-rack repair bandwidth γcsubscript𝛾𝑐\gamma_{c}, under the setting of n=15,k=8,L=3,=8formulae-sequence𝑛15formulae-sequence𝑘8formulae-sequence𝐿38n=15,k=8,L=3,\mathcal{M}=8

An example for the trade-off results of Corollary 1 is illustrated in Fig. 7, for various 0ϵ10italic-ϵ10\leq\epsilon\leq 1 values. Here, the ϵ=1italic-ϵ1\epsilon=1 (i.e., βI=βcsubscript𝛽𝐼subscript𝛽𝑐\beta_{I}=\beta_{c}) case corresponds to the symmetric repair in the non-clustered scenario [10]. The plot for ϵ=0italic-ϵ0\epsilon=0 (or βc=γc=0subscript𝛽𝑐subscript𝛾𝑐0\beta_{c}=\gamma_{c}=0) shows that the cross-cluster repair bandwidth can be reduced to zero with extra resources (α𝛼\alpha or γIsubscript𝛾𝐼\gamma_{I}), where the amount of required resources are specified in Corollary 1. Note that in the case of ϵ=0italic-ϵ0\epsilon=0, the storage system is completely localized, i.e., nodes can be repaired exclusively from their cluster. From Fig. 7, we can confirm that as ϵitalic-ϵ\epsilon decreases, extra resources (γ𝛾\gamma or α𝛼\alpha) are required to reliably store the given data \mathcal{M}. Moreover, Corollary 1 suggests a mathematically interesting result, stated in the following Theorem, the proof of which is in Appendix B.

Theorem 3.

A clustered DSS can reliably store file \mathcal{M} with the minimum storage overhead α=/k𝛼𝑘\alpha=\mathcal{M}/k if and only if

ϵ1nk.italic-ϵ1𝑛𝑘\epsilon\geq\frac{1}{n-k}. (27)

Note that α=/k𝛼𝑘\alpha=\mathcal{M}/k is the minimum storage overhead which can satisfy the MDS property, as stated in [10]. The implication of Theorem 3 is shown in Fig. 7. Under the =k=8𝑘8\mathcal{M}=k=8 setting, data \mathcal{M} can be reliably stored with minimum storage overhead α=/k=1𝛼𝑘1\alpha=\mathcal{M}/k=1 for 1nkϵ11𝑛𝑘italic-ϵ1\frac{1}{n-k}\leq\epsilon\leq 1, while it is impossible to achieve minimum storage overhead α=/k=1𝛼𝑘1\alpha=\mathcal{M}/k=1 for 0ϵ<1nk0italic-ϵ1𝑛𝑘0\leq\epsilon<\frac{1}{n-k}. Finally, since reducing the cross-cluster repair burden γcsubscript𝛾𝑐\gamma_{c} is regarded as a much harder problem compared to reducing the intra-cluster repair burden γIsubscript𝛾𝐼\gamma_{I}, we also plotted feasible (α,γc)𝛼subscript𝛾𝑐(\alpha,\gamma_{c}) pairs for various ϵitalic-ϵ\epsilon values in Fig. 8. The plot for ϵ=0italic-ϵ0\epsilon=0 obviously has zero γcsubscript𝛾𝑐\gamma_{c}, while ϵ>0italic-ϵ0\epsilon>0 cases show a trade-off relationship. As ϵitalic-ϵ\epsilon increases, the minimum γcsubscript𝛾𝑐\gamma_{c} value increases gradually.

IV-B Minimum-Bandwidth-Regenerating (MBR) point and Minimum-Storage-Regenerating (MSR) point

Refer to caption
Figure 9: Set of Feasible (α,γ)𝛼𝛾(\alpha,\gamma) Points

According to Corollary 1, the set of feasible (α,γ)𝛼𝛾(\alpha,\gamma) points shows a trade-off curve as in Fig. 9, for arbitrary n,k,L,ϵ𝑛𝑘𝐿italic-ϵn,k,L,\epsilon settings. Here we focus on two extremal points: the minimum-bandwidth-regenerating (MBR) point and the minimum-storage-regenerating (MSR) point. As originally defined in [10], we call the point on the trade-off with minimum bandwidth γ𝛾\gamma as MBR. Similarly, we call the point with minimum storage α𝛼\alpha as MSR222Among multiple points with minimum γ𝛾\gamma, the point having the smallest α𝛼\alpha is called the MBR point. Similarly, among points with minimum α𝛼\alpha, the point with the minimum γ𝛾\gamma is called the MSR point.. Let (αmsr(ϵ),γmsr(ϵ))superscriptsubscript𝛼𝑚𝑠𝑟italic-ϵsuperscriptsubscript𝛾𝑚𝑠𝑟italic-ϵ(\alpha_{msr}^{(\epsilon)},\gamma_{msr}^{(\epsilon)}) be the (α,γ)𝛼𝛾(\alpha,\gamma) pair of the MSR point for given ϵitalic-ϵ\epsilon. Similarly, define (αmbr(ϵ),γmbr(ϵ))superscriptsubscript𝛼𝑚𝑏𝑟italic-ϵsuperscriptsubscript𝛾𝑚𝑏𝑟italic-ϵ(\alpha_{mbr}^{(\epsilon)},\gamma_{mbr}^{(\epsilon)}) as the parameter pair for MBR points. According to Corollary 1, the explicit (α,γ)𝛼𝛾(\alpha,\gamma) expression for the MSR and MBR points are as in the following Corollary, the proof of which is given in Appendix F-B.

Corollary 2.

For a given 0ϵ10italic-ϵ10\leq\epsilon\leq 1, we have

(αmsr(ϵ),γmsr(ϵ))superscriptsubscript𝛼𝑚𝑠𝑟italic-ϵsuperscriptsubscript𝛾𝑚𝑠𝑟italic-ϵ\displaystyle(\alpha_{msr}^{(\epsilon)},\gamma_{msr}^{(\epsilon)})
={(τ+i=τ+1kzi,τ+i=τ+1kzii=τ+1kzisτ),0ϵ<1nk(k,k1sk1),1nkϵ1absentcases𝜏superscriptsubscript𝑖𝜏1𝑘subscript𝑧𝑖𝜏superscriptsubscript𝑖𝜏1𝑘subscript𝑧𝑖superscriptsubscript𝑖𝜏1𝑘subscript𝑧𝑖subscript𝑠𝜏0italic-ϵ1𝑛𝑘𝑘𝑘1subscript𝑠𝑘11𝑛𝑘italic-ϵ1\displaystyle=\begin{cases}(\frac{\mathcal{M}}{\tau+\sum_{i=\tau+1}^{k}z_{i}},\frac{\mathcal{M}}{\tau+\sum_{i=\tau+1}^{k}z_{i}}\frac{\sum_{i=\tau+1}^{k}z_{i}}{s_{\tau}}),&0\leq\epsilon<\frac{1}{n-k}\\ (\frac{\mathcal{M}}{k},\frac{\mathcal{M}}{k}\frac{1}{s_{k-1}}),&\frac{1}{n-k}\leq\epsilon\leq 1\end{cases} (28)
(αmbr(ϵ),γmbr(ϵ))superscriptsubscript𝛼𝑚𝑏𝑟italic-ϵsuperscriptsubscript𝛾𝑚𝑏𝑟italic-ϵ\displaystyle(\alpha_{mbr}^{(\epsilon)},\gamma_{mbr}^{(\epsilon)}) =(s0,s0).absentsubscript𝑠0subscript𝑠0\displaystyle=(\frac{\mathcal{M}}{s_{0}},\frac{\mathcal{M}}{s_{0}}). (29)

Now we compare the MSR and MBR points for two extreme cases of ϵ=0italic-ϵ0\epsilon=0 and ϵ=1italic-ϵ1\epsilon=1. Using Rk/n𝑅𝑘𝑛R\coloneqq k/n and the dispersion ratio σ𝜎\sigma defined in (13), the asymptotic behaviors of MBR and MSR points are illustrated in the following theorem, the proof of which is in Appendix C.

Theorem 4.

Consider the MSR point (αmsr(ϵ),γmsr(ϵ))superscriptsubscript𝛼𝑚𝑠𝑟italic-ϵsuperscriptsubscript𝛾𝑚𝑠𝑟italic-ϵ(\alpha_{msr}^{(\epsilon)},\gamma_{msr}^{(\epsilon)}) and the MBR point (αmbr(ϵ),γmbr(ϵ))superscriptsubscript𝛼𝑚𝑏𝑟italic-ϵsuperscriptsubscript𝛾𝑚𝑏𝑟italic-ϵ(\alpha_{mbr}^{(\epsilon)},\gamma_{mbr}^{(\epsilon)}) for ϵ=0,1italic-ϵ01\epsilon=0,1. The minimum node storage for ϵ=0italic-ϵ0\epsilon=0 is asymptotically equivalent to the minimum node storage for ϵ=1italic-ϵ1\epsilon=1, i.e.,

αmsr(0)αmsr(1)=/ksimilar-tosuperscriptsubscript𝛼𝑚𝑠𝑟0superscriptsubscript𝛼𝑚𝑠𝑟1𝑘\alpha_{msr}^{(0)}\sim\alpha_{msr}^{(1)}=\mathcal{M}/k (30)

as n𝑛n\rightarrow\infty for arbitrary fixed σ𝜎\sigma and R=k/n𝑅𝑘𝑛R=k/n. Moreover, the MBR point for ϵ=0italic-ϵ0\epsilon=0 approaches the MBR point for ϵ=1italic-ϵ1\epsilon=1, i.e.,

(αmbr(0),γmbr(0))(αmbr(1),γmbr(1)),superscriptsubscript𝛼𝑚𝑏𝑟0superscriptsubscript𝛾𝑚𝑏𝑟0superscriptsubscript𝛼𝑚𝑏𝑟1superscriptsubscript𝛾𝑚𝑏𝑟1(\alpha_{mbr}^{(0)},\gamma_{mbr}^{(0)})\rightarrow(\alpha_{mbr}^{(1)},\gamma_{mbr}^{(1)}), (31)

as R=k/n1𝑅𝑘𝑛1R=k/n\rightarrow 1. The ratio between γmbr(0)superscriptsubscript𝛾𝑚𝑏𝑟0\gamma_{mbr}^{(0)} and γmbr(1)superscriptsubscript𝛾𝑚𝑏𝑟1\gamma_{mbr}^{(1)} is expressed as

γmbr(0)γmbr(1)2k1n1superscriptsubscript𝛾𝑚𝑏𝑟0superscriptsubscript𝛾𝑚𝑏𝑟12𝑘1𝑛1\frac{\gamma_{mbr}^{(0)}}{\gamma_{mbr}^{(1)}}\leq 2-\frac{k-1}{n-1} (32)
Refer to caption
Figure 10: Optimal trade-off curves for ϵ=0,1italic-ϵ01\epsilon=0,1

Note that under the setting of Theorem 4, we have

αmsr(1)superscriptsubscript𝛼𝑚𝑠𝑟1\displaystyle\alpha_{msr}^{(1)} =constant,k=nR=Θ(n),=Θ(n)formulae-sequenceformulae-sequenceabsentconstant𝑘𝑛𝑅Θ𝑛Θ𝑛\displaystyle=\text{constant},\quad k=nR=\Theta(n),\quad\mathcal{M}=\Theta(n)

in the asymptotic regime of large n𝑛n. According to Theorem 4, the minimum storage for ϵ=0italic-ϵ0\epsilon=0 can achieve /k𝑘\mathcal{M}/k as n𝑛n\rightarrow\infty with fixed R=k/n𝑅𝑘𝑛R=k/n. This result coincides with the result of Theorem 3. According to Theorem 3, the sufficient and necessary condition for achieving the minimum storage of α=/k𝛼𝑘\alpha=\mathcal{M}/k is

ϵ1nk=1n(1R).italic-ϵ1𝑛𝑘1𝑛1𝑅\epsilon\geq\frac{1}{n-k}=\frac{1}{n(1-R)}. (33)

As n𝑛n increases with a fixed R𝑅R, the lower bound on ϵitalic-ϵ\epsilon reduces, so that in the asymptotic regime, ϵ=0italic-ϵ0\epsilon=0 can achieve α=/k𝛼𝑘\alpha=\mathcal{M}/k.

Refer to caption
(a)
Refer to caption
(b)
Figure 11: MBR coding schemes for n=6,k=5,L=2,α=10formulae-sequence𝑛6formulae-sequence𝑘5formulae-sequence𝐿2𝛼10n=6,k=5,L=2,\alpha=10 [sub-symbol], γ=10𝛾10\gamma=10 [sub-symbol]

Moreover, Theorem 4 states that the MBR point for ϵ=0italic-ϵ0\epsilon=0 approaches the MBR point for ϵ=1italic-ϵ1\epsilon=1 as R=k/n𝑅𝑘𝑛R=k/n goes to 1. Fig. 11 provides two MBR coding schemes with (n,k,L)=(6,5,2)𝑛𝑘𝐿652(n,k,L)=(6,5,2), which has different ϵitalic-ϵ\epsilon values; one coding scheme in Fig. 11a satisfies ϵ=1italic-ϵ1\epsilon=1, while the other in Fig. 11b satisfies ϵ=0italic-ϵ0\epsilon=0. The RSKR coding scheme [12] is applied to the six nodes in Fig. 11a. Each node (illustrated as a rectangular box) contains five cisubscript𝑐𝑖c_{i} symbols, where each symbol cisubscript𝑐𝑖c_{i} consists of two sub-symbols, ci(1)superscriptsubscript𝑐𝑖1c_{i}^{(1)} and ci(2)superscriptsubscript𝑐𝑖2c_{i}^{(2)}. Note that any symbol cisubscript𝑐𝑖c_{i} is shared by exactly two nodes in Fig. 11a, which is due to the property of RSKR coding. This system can reliably store fifteen symbols {ci}i=115superscriptsubscriptsubscript𝑐𝑖𝑖115\{c_{i}\}_{i=1}^{15}, or =3030\mathcal{M}=30 sub-symbols {ci(1),ci(2)}i=115superscriptsubscriptsuperscriptsubscript𝑐𝑖1superscriptsubscript𝑐𝑖2𝑖115\{c_{i}^{(1)},c_{i}^{(2)}\}_{i=1}^{15}, since it satisfies two properties – the exact repair property and the data recovery property – as illustrated below. First, when a node fails, five other nodes transmit five symbols (one distinct symbol by each node), which exactly regenerates the failed node. Second, we can retrieve data, =3030\mathcal{M}=30 sub-symbols {ci(1),ci(2)}i=115superscriptsubscriptsuperscriptsubscript𝑐𝑖1superscriptsubscript𝑐𝑖2𝑖115\{c_{i}^{(1)},c_{i}^{(2)}\}_{i=1}^{15}, by contacting any k=5𝑘5k=5 nodes. In Fig. 11b, each node contains two eisubscript𝑒𝑖e_{i} symbols, where each symbol eisubscript𝑒𝑖e_{i} consists of five sub-symbols, {ei(j)}j=15superscriptsubscriptsuperscriptsubscript𝑒𝑖𝑗𝑗15\{e_{i}^{(j)}\}_{j=1}^{5}. Note that in Fig. 11b, any symbol eisubscript𝑒𝑖e_{i} is shared by exactly two nodes which reside in the same cluster. This is because we applied RSKR coding at each cluster in the system of Fig. 11b. This system can reliably store six symbols {ei}i=16superscriptsubscriptsubscript𝑒𝑖𝑖16\{e_{i}\}_{i=1}^{6}, or =3030\mathcal{M}=30 sub-symbols i[6],j[5]{ei(j)}subscriptformulae-sequence𝑖delimited-[]6𝑗delimited-[]5superscriptsubscript𝑒𝑖𝑗\bigcup_{i\in[6],j\in[5]}\{e_{i}^{(j)}\}, since it satisfies the exact repair property and the data recovery property.

Note that both DSSs in Fig. 11 reliably store =3030\mathcal{M}=30 sub-symbols, by using the node capacity of α=10𝛼10\alpha=10 sub-symbols and the repair bandwidth of γ=10𝛾10\gamma=10 sub-symbols. However, the former system requires γc=6subscript𝛾𝑐6\gamma_{c}=6 cross-cluster repair bandwidth for each node failure event, while the latter system requires γc=0subscript𝛾𝑐0\gamma_{c}=0 cross-cluster repair bandwidth. For example, if the leftmost node of the 1stsuperscript1𝑠𝑡1^{st} cluster fails in Fig. 11a, then four sub-symbols {c1(i),c2(i)}i=12superscriptsubscriptsuperscriptsubscript𝑐1𝑖superscriptsubscript𝑐2𝑖𝑖12\{c_{1}^{(i)},c_{2}^{(i)}\}_{i=1}^{2} are transmitted within that cluster, while six sub-symbols {c3(i),c4(i),c5(i)}i=12superscriptsubscriptsuperscriptsubscript𝑐3𝑖superscriptsubscript𝑐4𝑖superscriptsubscript𝑐5𝑖𝑖12\{c_{3}^{(i)},c_{4}^{(i)},c_{5}^{(i)}\}_{i=1}^{2} are transmitted from the 2ndsuperscript2𝑛𝑑2^{nd} cluster. In the case of ϵ=0italic-ϵ0\epsilon=0 in Fig. 11b, ten sub-symbols {e1(i),e2(i)}i=15superscriptsubscriptsuperscriptsubscript𝑒1𝑖superscriptsubscript𝑒2𝑖𝑖15\{e_{1}^{(i)},e_{2}^{(i)}\}_{i=1}^{5} are transmitted within the 1stsuperscript1𝑠𝑡1^{st} cluster and no sub-symbols are transmitted across the clusters, when the leftmost node of the 1stsuperscript1𝑠𝑡1^{st} cluster fails. Thus, transition from the former system (Fig. 11a) to the latter system (Fig. 11b) reduces the cross-cluster repair bandwidth to zero, while maintaining the storage capacity \mathcal{M} and the required resource pair (α,γ)𝛼𝛾(\alpha,\gamma). Likewise, we can reduce the cross-cluster repair bandwidth to zero while maintaining the storage capacity, in the case of R=k/n1𝑅𝑘𝑛1R=k/n\rightarrow 1.

Note that γmbr(ϵ)=αmbr(ϵ)superscriptsubscript𝛾𝑚𝑏𝑟italic-ϵsuperscriptsubscript𝛼𝑚𝑏𝑟italic-ϵ\gamma_{mbr}^{(\epsilon)}=\alpha_{mbr}^{(\epsilon)} for 0ϵ10italic-ϵ10\leq\epsilon\leq 1, from (29). Thus, the result of (32) in Theorem 4 can be expressed as Fig. 12 at the asymptotic regime of large n,k𝑛𝑘n,k. According to Fig. 12, intra-cluster only repair (ϵ=0italic-ϵ0\epsilon=0 or βc=0subscript𝛽𝑐0\beta_{c}=0) is possible by using additional resources (α𝛼\alpha and γ𝛾\gamma) in the 1R1𝑅1-R portion, compared to the symmetric repair (ϵ=1italic-ϵ1\epsilon=1) case.

Refer to caption
Figure 12: Relationship between MBR point of ϵ=0italic-ϵ0\epsilon=0 and that of ϵ=1italic-ϵ1\epsilon=1

IV-C Intra-cluster Repairable Codes versus Locally Repairable Codes [32]

Here, we define a family of network coding schemes for clustered DSS, which we call the intra-cluster repairable codes. In Corollary 1, we considered DSSs with ϵ=0italic-ϵ0\epsilon=0, which can repair any failed node by using intra-cluster communication only. The optimal (α,γ)𝛼𝛾(\alpha,\gamma) trade-off curve which satisfies ϵ=0italic-ϵ0\epsilon=0 is illustrated as the solid line with cross markers in Fig. 7. Each point on the curve is achievable (i.e., there exists a network coding scheme), according to the result of [11]. We call the network coding schemes for the points on the curve of ϵ=0italic-ϵ0\epsilon=0 the intra-cluster repairable codes, since these coding schemes can repair any failed node by using intra-cluster communication only. The relationship between the intra-cluster repairable codes and the locally repairable codes (LRC) of [32] are investigated in Theorem 5, the proof of which is given in Appendix D. Note that according to the definition in [32], an (n,l0,m0,,α)𝑛subscript𝑙0subscript𝑚0𝛼(n,l_{0},m_{0},\mathcal{M},\alpha)-LRC encodes a file of size \mathcal{M} into n𝑛n coded symbols, where each symbol contains α𝛼\alpha bits. In addition, any coded symbol of the LRC is regenerated by accessing at most l0subscript𝑙0l_{0} other symbols (i.e., the code has repair locality of l0subscript𝑙0l_{0}), while the minimum distance of the code is m0subscript𝑚0m_{0}.

Theorem 5.

The intra-cluster repairable codes with storage overhead α𝛼\alpha are the (n,l0,m0,,α)𝑛subscript𝑙0subscript𝑚0𝛼(n,l_{0},m_{0},\mathcal{M},\alpha)-LRC codes of [32] where

l0subscript𝑙0\displaystyle l_{0} =nI1,absentsubscript𝑛𝐼1\displaystyle=n_{I}-1,
m0subscript𝑚0\displaystyle m_{0} =nk+1.absent𝑛𝑘1\displaystyle=n-k+1.

It is confirmed that Theorem 1 of [32],

m0nαl0α+2,subscript𝑚0𝑛𝛼subscript𝑙0𝛼2m_{0}\leq n-\bigg{\lceil}\frac{\mathcal{M}}{\alpha}\bigg{\rceil}-\bigg{\lceil}\frac{\mathcal{M}}{l_{0}\alpha}\bigg{\rceil}+2, (34)

holds for every intra-cluster repairable code with storage overhead α𝛼\alpha. Moreover, the equality of (34) holds if

α=αmsr(0),(kmodnI)0.formulae-sequence𝛼superscriptsubscript𝛼𝑚𝑠𝑟0𝑘modsubscript𝑛𝐼0\displaystyle\alpha=\alpha_{msr}^{(0)},(k\ \mathrm{mod}\ n_{I})\neq 0.

IV-D Required βcsubscript𝛽𝑐\beta_{c} for a given α𝛼\alpha

Here we focus on the following question: when the available intra-cluster repair bandwidth is abundant, how much cross-cluster repair bandwidth is required to reliably store file \mathcal{M}? We consider scenarios when the intra-cluster repair bandwidth (per node) has its maximum value, i.e., βI=αsubscript𝛽𝐼𝛼\beta_{I}=\alpha. Under this setting, Theorem 6 specifies the minimum required βcsubscript𝛽𝑐\beta_{c} which satisfies 𝒞(α,βI,βc)𝒞𝛼subscript𝛽𝐼subscript𝛽𝑐\mathcal{C}(\alpha,\beta_{I},\beta_{c})\geq\mathcal{M}. The proof of Theorem 6 is given in Appendix E.

Theorem 6.

Suppose the intra-cluster repair bandwidth is set at the maximum value, i.e., βI=αsubscript𝛽𝐼𝛼\beta_{I}=\alpha. For a given node capacity α𝛼\alpha, the clustered DSS can reliably store data \mathcal{M} if and only if βcβcsubscript𝛽𝑐superscriptsubscript𝛽𝑐\beta_{c}\geq\beta_{c}^{*} where

βcsuperscriptsubscript𝛽𝑐\displaystyle\beta_{c}^{*} ={(k1)αnk, if α[k,fk1)mαi=m+1k(ni), if α[fm+1,fm)(m=k2,k3,,s+1)k0αi=k0+1k(ni), if α[fk0+1,k0)0, if α[k0,),absentcases𝑘1𝛼𝑛𝑘 if 𝛼𝑘subscript𝑓𝑘1𝑚𝛼superscriptsubscript𝑖𝑚1𝑘𝑛𝑖 if 𝛼subscript𝑓𝑚1subscript𝑓𝑚otherwise𝑚𝑘2𝑘3𝑠1subscript𝑘0𝛼superscriptsubscript𝑖subscript𝑘01𝑘𝑛𝑖 if 𝛼subscript𝑓subscript𝑘01subscript𝑘00 if 𝛼subscript𝑘0\displaystyle=\begin{cases}\frac{\mathcal{M}-(k-1)\alpha}{n-k},&\text{ if }\alpha\in[\frac{\mathcal{M}}{k},\frac{\mathcal{M}}{f_{k-1}})\\ \frac{\mathcal{M}-m\alpha}{\sum_{i=m+1}^{k}(n-i)},&\text{ if }\alpha\in[\frac{\mathcal{M}}{f_{m+1}},\frac{\mathcal{M}}{f_{m}})\\ &\quad\quad(m=k-2,k-3,\cdots,s+1)\\ \frac{\mathcal{M}-k_{0}\alpha}{\sum_{i=k_{0}+1}^{k}(n-i)},&\text{ if }\alpha\in[\frac{\mathcal{M}}{f_{k_{0}+1}},\frac{\mathcal{M}}{k_{0}})\\ 0,&\text{ if }\alpha\in[\frac{\mathcal{M}}{k_{0}},\infty),\end{cases} (35)
fmsubscript𝑓𝑚\displaystyle f_{m} =m+i=m+1k(ni)nm,absent𝑚superscriptsubscript𝑖𝑚1𝑘𝑛𝑖𝑛𝑚\displaystyle=m+\frac{\sum_{i=m+1}^{k}(n-i)}{n-m}, (36)
k0subscript𝑘0\displaystyle k_{0} =kknI.absent𝑘𝑘subscript𝑛𝐼\displaystyle=k-\lfloor\frac{k}{n_{I}}\rfloor. (37)
Refer to caption
Figure 13: Optimal tradeoff between cross-cluster repair bandwidth and node capacity, when n=100,k=85,L=10,=85formulae-sequence𝑛100formulae-sequence𝑘85formulae-sequence𝐿1085n=100,k=85,L=10,\mathcal{M}=85 and βI=αsubscript𝛽𝐼𝛼\beta_{I}=\alpha

Fig. 13 provides an example of the optimal trade-off relationship between βcsubscript𝛽𝑐\beta_{c} and α𝛼\alpha, explained in Theorem 6. For α/k0=1.104𝛼subscript𝑘01.104\alpha\geq\mathcal{M}/k_{0}=1.104, the cross-cluster burden βcsubscript𝛽𝑐\beta_{c} can be reduced to zero. However, as α𝛼\alpha decreases from /k0subscript𝑘0\mathcal{M}/k_{0}, the system requires a larger βcsubscript𝛽𝑐\beta_{c} value. For example, if α=βI=1.05𝛼subscript𝛽𝐼1.05\alpha=\beta_{I}=1.05 in Fig. 13, βc0.03subscript𝛽𝑐0.03\beta_{c}\geq 0.03 is required to satisfy 𝒞(α,βI,βc)=85𝒞𝛼subscript𝛽𝐼subscript𝛽𝑐85\mathcal{C}(\alpha,\beta_{I},\beta_{c})\geq\mathcal{M}=85. Thus, for each node failure event, a cross-cluster repair bandwidth of γc=(nnI)βc2.7subscript𝛾𝑐𝑛subscript𝑛𝐼subscript𝛽𝑐2.7\gamma_{c}=(n-n_{I})\beta_{c}\geq 2.7 is required. Theorem 6 provides an explicit equation for the cross-cluster repair bandwidth we need to pay, in order to reduce node capacity α𝛼\alpha.

V Further Comments & Future Works

V-A Explicit coding schemes for clustered DSS

According to the part I proof of Theorem 1, there exists an information flow graph Gsuperscript𝐺G^{*} which has the min-cut value of 𝒞𝒞\mathcal{C}, the capacity of clustered DSS. Thus, according to [11], there exists a linear network coding scheme which achieves capacity 𝒞𝒞\mathcal{C}. Although the existence of a coding scheme is verified, explicit network coding schemes which achieve capacity need to be specified for implementing practical systems. Recently, under the setting of clustered DSS modeled in the present paper, MBR codes for all n,k,L,ϵ𝑛𝑘𝐿italic-ϵn,k,L,\epsilon are constructed in [36] and MSR codes for limited parameters are designed in [37]. Explicit code construction for general parameters and/or construction of codes that requires small field size are interesting remaining issues.

V-B Optimal number of clusters

According to Theorem 2, capacity 𝒞𝒞\mathcal{C} is asymptotically a monotonically decreasing function of L𝐿L, the number of clusters. Thus, reducing the number of clusters (i.e., gathering storage nodes into a smaller number of clusters) increases storage capacity. However, as mentioned in Section III-E, we typically want to have a sufficiently large L𝐿L, to tolerate the failure of a cluster. Then, the remaining question is in finding optimal Lsuperscript𝐿L^{*} which not only allows sufficiently large storage capacity, but also a tolerance to cluster failures. We regard this problem as a future research topic, the solution to which will provide a guidance on the strategy for distributing storage nodes into multiple clusters.

V-C Extension to general dI,dcsubscript𝑑𝐼subscript𝑑𝑐d_{I},d_{c} settings

The present paper assumed a maximum helper node setting, dI=nI1subscript𝑑𝐼subscript𝑛𝐼1d_{I}=n_{I}-1 and dc=nnIsubscript𝑑𝑐𝑛subscript𝑛𝐼d_{c}=n-n_{I}, since it maximizes the capacity as stated in Proposition 1. However, waiting for all helper nodes gives rise to a latency issue. If we reduce the number of helper nodes dIsubscript𝑑𝐼d_{I} and dcsubscript𝑑𝑐d_{c}, low latency repair would be possible, while the achievable storage capacity decreases. Thus, we consider obtaining the capacity expression for general dI,dcsubscript𝑑𝐼subscript𝑑𝑐d_{I},d_{c} settings, and discover the trade-off between capacity and latency for various dI,dcsubscript𝑑𝐼subscript𝑑𝑐d_{I},d_{c} values.

V-D Scenarios of aggregating the helper data within each cluster

In recent work [28], [38] on multi-rack (multi-cluster) distributed storage, the authors discuss aggregation of repair data leaving a given cluster. The ideas is to allow aggregation and compression of all helper data leaving each cluster to aid reconstruction taking place in some other cluster containing a failed node. This type of repair link aggregation has been shown to reduce the cross-cluster repair burden in [28], [38]. We expect that the same method would also change the tradeoff picture for our distributed cluster model. This is certainly an interesting and important topic to investigate, but careful analysis including the effect of security breach on links will provide a more complete assessment of the merits and potential perils of repair link aggregation. We will leave this as a future endeavor.

VI Conclusion

This paper considered a practical distributed storage system where storage nodes are dispersed into several clusters. Noticing that the traffic burdens of intra- and cross-cluster communications are typically different, a new system model for clustered distributed storage systems is suggested. Based on the cut-set bound analysis of information flow graph, the storage capacity 𝒞(α,βI,βc)𝒞𝛼subscript𝛽𝐼subscript𝛽𝑐\mathcal{C}(\alpha,\beta_{I},\beta_{c}) of the suggested model is obtained in a closed-form, as a function of three main resources: node storage capacity α𝛼\alpha, intra-cluster repair bandwidth βIsubscript𝛽𝐼\beta_{I} and cross-cluster repair bandwidth βcsubscript𝛽𝑐\beta_{c}. It is shown that the asymmetric repair (βI>βcsubscript𝛽𝐼subscript𝛽𝑐\beta_{I}>\beta_{c}) degrades capacity, which is the cost for lifting the cross-cluster repair burden. Moreover, in the asymptotic regime of a large number of storage nodes, capacity is shown to be asymptotically equivalent to a monotonic decreasing function of L𝐿L, the number of clusters. Thus, reducing L𝐿L (i.e., gathering nodes into less clusters) is beneficial for increasing capacity, although we would typically need to guarantee sufficiently large L𝐿L to tolerate rack failure events.

Using the capacity expression, we obtained the feasible set of (α,βI,βc𝛼subscript𝛽𝐼subscript𝛽𝑐\alpha,\beta_{I},\beta_{c}) triplet which satisfies 𝒞(α,βI,βc)𝒞𝛼subscript𝛽𝐼subscript𝛽𝑐\mathcal{C}(\alpha,\beta_{I},\beta_{c})\geq\mathcal{M}, i.e., it is possible to reliably store file \mathcal{M} by using the resource value set (α,βI,βc𝛼subscript𝛽𝐼subscript𝛽𝑐\alpha,\beta_{I},\beta_{c}). The closed-form solution on the feasible set shows a different behavior depending on ϵ=βc/βIitalic-ϵsubscript𝛽𝑐subscript𝛽𝐼\epsilon=\beta_{c}/\beta_{I}, the ratio of cross- to intra-cluster repair bandwidth. It is shown that the minimum storage of α=/k𝛼𝑘\alpha=\mathcal{M}/k is achievable if and only if ϵ1nkitalic-ϵ1𝑛𝑘\epsilon\geq\frac{1}{n-k}. Moreover, in the special case of ϵ=0italic-ϵ0\epsilon=0, we can construct a reliable storage system without using cross-cluster repair bandwidth. A family of network codes which enable ϵ=0italic-ϵ0\epsilon=0, called the intra-cluster repairable codes, has been shown to be a class of the locally repairable codes defined in [32].

Appendix A Proof of Theorem 1

Here, we prove Theorem 1. First, denote the right-hand-side (RHS) of (5) as

Ti=1nIj=1gimin{α,ρiβI+(nρijm=1i1gm)βc}.𝑇superscriptsubscript𝑖1subscript𝑛𝐼superscriptsubscript𝑗1subscript𝑔𝑖𝛼subscript𝜌𝑖subscript𝛽𝐼𝑛subscript𝜌𝑖𝑗superscriptsubscript𝑚1𝑖1subscript𝑔𝑚subscript𝛽𝑐T\coloneqq\sum_{i=1}^{n_{I}}\sum_{j=1}^{g_{i}}\min\{\alpha,\rho_{i}\beta_{I}+(n-\rho_{i}-j-\sum_{m=1}^{i-1}g_{m})\beta_{c}\}. (A.1)

For other notations used in this proof, refer to subsection II-C. The proof proceeds in two parts.

Part I. Show an information flow graph G𝒢superscript𝐺𝒢G^{*}\in\mathcal{G} and a cut-set cC(G)superscript𝑐𝐶superscript𝐺c^{*}\in C(G^{*}) such that w(G,c)=T𝑤superscript𝐺superscript𝑐𝑇w(G^{*},c^{*})=T:

Refer to caption
Figure 14: The information flow graph Gsuperscript𝐺G^{*} for the part I proof of Theorem 1

Consider the information flow graph Gsuperscript𝐺G^{*} illustrated in Fig. 14, which is obtained by the following procedure. First, data from source node S𝑆S is distributed into n𝑛n nodes labeled from x1superscript𝑥1x^{1} to xnsuperscript𝑥𝑛x^{n}. As mentioned in Section II-B, the storage node xi=(xini,xouti)superscript𝑥𝑖subscriptsuperscript𝑥𝑖𝑖𝑛subscriptsuperscript𝑥𝑖𝑜𝑢𝑡x^{i}=(x^{i}_{in},x^{i}_{out}) consists of an input-node xinisuperscriptsubscript𝑥𝑖𝑛𝑖x_{in}^{i} and an output-node xoutisuperscriptsubscript𝑥𝑜𝑢𝑡𝑖x_{out}^{i}. Second, storage node xtsuperscript𝑥𝑡x^{t} fails and is regenerated at the newcomer node xn+tsuperscript𝑥𝑛𝑡x^{n+t} for t[k]𝑡delimited-[]𝑘t\in[k]. The newcomer node xn+tsuperscript𝑥𝑛𝑡x^{n+t} connects to n1𝑛1n-1 survived nodes {xm}m=t+1n+t1superscriptsubscriptsuperscript𝑥𝑚𝑚𝑡1𝑛𝑡1\{x^{m}\}_{m=t+1}^{n+t-1} to regenerate xtsuperscript𝑥𝑡x^{t}. Third, data collector node DC𝐷𝐶DC contacts {xn+t}t=1ksuperscriptsubscriptsuperscript𝑥𝑛𝑡𝑡1𝑘\{x^{n+t}\}_{t=1}^{k} to retrieve data. This whole process is illustrated in the information flow graph Gsuperscript𝐺G^{*}.

Refer to caption
Figure 15: 2-dim. structure representation

To specify Gsuperscript𝐺G^{*}, here we determine the 2-dimensional location of the k𝑘k newcomer nodes {xn+t}t=1ksuperscriptsubscriptsuperscript𝑥𝑛𝑡𝑡1𝑘\{x^{n+t}\}_{t=1}^{k}. First, consider the 2-dimensional structure representation of clustered distributed storage, illustrated in Fig. 15. In this figure, each row represents each cluster, and each node is represented as a 2-dimensional (i,j)𝑖𝑗(i,j) point for i[L]𝑖delimited-[]𝐿i\in[L] and j[nI]𝑗delimited-[]subscript𝑛𝐼j\in[n_{I}]. The symbol N(i,j)𝑁𝑖𝑗N(i,j) denotes the node at (i,j)𝑖𝑗(i,j) point. Here we define the set of n𝑛n nodes,

𝒩{N(i,j):i[L],j[nI]}.𝒩conditional-set𝑁𝑖𝑗formulae-sequence𝑖delimited-[]𝐿𝑗delimited-[]subscript𝑛𝐼\mathcal{N}\coloneqq\{N(i,j):i\in[L],j\in[n_{I}]\}. (A.2)

For t[k]𝑡delimited-[]𝑘t\in[k], consider selecting the newcomer node xn+tsuperscript𝑥𝑛𝑡x^{n+t} as

xn+t=N(it,jt)superscript𝑥𝑛𝑡𝑁subscript𝑖𝑡subscript𝑗𝑡x^{n+t}=N(i_{t},j_{t}) (A.3)

where

itsubscript𝑖𝑡\displaystyle i_{t} =min{ν[nI]:m=1νgmt},absent:𝜈delimited-[]subscript𝑛𝐼superscriptsubscript𝑚1𝜈subscript𝑔𝑚𝑡\displaystyle=\min\{\nu\in[n_{I}]:\sum_{m=1}^{\nu}g_{m}\geq t\}, (A.4)
jtsubscript𝑗𝑡\displaystyle j_{t} =tm=1it1gm,absent𝑡superscriptsubscript𝑚1subscript𝑖𝑡1subscript𝑔𝑚\displaystyle=t-\sum_{m=1}^{i_{t}-1}g_{m}, (A.5)

and gmsubscript𝑔𝑚g_{m} used in the method is defined in (7). The location of k𝑘k newcomer nodes selected by this method are illustrated in Fig. 16. Moreover, for the n=12,L=3,k=9formulae-sequence𝑛12formulae-sequence𝐿3𝑘9n=12,L=3,k=9 case, the newcomer nodes {xn+t}t=1ksuperscriptsubscriptsuperscript𝑥𝑛𝑡𝑡1𝑘\{x^{n+t}\}_{t=1}^{k} are depicted in Fig. 17. In these figures, the node with number t𝑡t inside represents the newcomer node labeled as xn+tsuperscript𝑥𝑛𝑡x^{n+t}.

Refer to caption
Figure 16: The location of k𝑘k newcomer nodes: xn+1,,xn+ksuperscript𝑥𝑛1superscript𝑥𝑛𝑘x^{n+1},\cdots,x^{n+k}
Refer to caption
Figure 17: The location of k𝑘k newcomer nodes for n=12,L=3,k=9formulae-sequence𝑛12formulae-sequence𝐿3𝑘9n=12,L=3,k=9 case

For the given graph Gsuperscript𝐺G^{*}, now we consider a cut-set cC(G)superscript𝑐𝐶superscript𝐺c^{*}\in C(G^{*}) defined as below. The cut-set c=(U,U¯)superscript𝑐𝑈¯𝑈c^{*}=(U,\mkern 1.5mu\overline{\mkern-1.5muU\mkern-1.5mu}\mkern 1.5mu) can be defined by specifying U𝑈U and U¯¯𝑈\mkern 1.5mu\overline{\mkern-1.5muU\mkern-1.5mu}\mkern 1.5mu (complement of U𝑈U), which partition the set of vertices in Gsuperscript𝐺G^{*}. First, let xini,xoutiUsubscriptsuperscript𝑥𝑖𝑖𝑛subscriptsuperscript𝑥𝑖𝑜𝑢𝑡𝑈x^{i}_{in},x^{i}_{out}\in U for i[n]𝑖delimited-[]𝑛i\in[n] and xoutn+iU¯subscriptsuperscript𝑥𝑛𝑖𝑜𝑢𝑡¯𝑈x^{n+i}_{out}\in\mkern 1.5mu\overline{\mkern-1.5muU\mkern-1.5mu}\mkern 1.5mu for i[k]𝑖delimited-[]𝑘i\in[k]. For i[k]𝑖delimited-[]𝑘i\in[k], the input node xinn+isubscriptsuperscript𝑥𝑛𝑖𝑖𝑛x^{n+i}_{in} is included in either U𝑈U or U¯¯𝑈\mkern 1.5mu\overline{\mkern-1.5muU\mkern-1.5mu}\mkern 1.5mu, depending on the condition specified in the next paragraph. Moreover, let SU𝑆𝑈S\in U and DCU¯𝐷𝐶¯𝑈DC\in\mkern 1.5mu\overline{\mkern-1.5muU\mkern-1.5mu}\mkern 1.5mu. See Fig. 18.

Let U0=i=1n{xouti}subscript𝑈0superscriptsubscript𝑖1𝑛subscriptsuperscript𝑥𝑖𝑜𝑢𝑡U_{0}=\bigcup_{i=1}^{n}\{x^{i}_{out}\}. For t[k]𝑡delimited-[]𝑘t\in[k], let ωtsuperscriptsubscript𝜔𝑡\omega_{t}^{*} be the sum of capacities of edges from U0subscript𝑈0U_{0} to xinn+tsubscriptsuperscript𝑥𝑛𝑡𝑖𝑛x^{n+t}_{in}. If αωt𝛼superscriptsubscript𝜔𝑡\alpha\leq\omega_{t}^{*}, then we include xinn+tsuperscriptsubscript𝑥𝑖𝑛𝑛𝑡x_{in}^{n+t} in U𝑈U. Otherwise, we include xinn+tsubscriptsuperscript𝑥𝑛𝑡𝑖𝑛x^{n+t}_{in} in U¯¯𝑈\mkern 1.5mu\overline{\mkern-1.5muU\mkern-1.5mu}\mkern 1.5mu. Then, the cut-set csuperscript𝑐c^{*} has the cut-value of

w(G,c)=t=1kmin{α,ωt}.𝑤superscript𝐺superscript𝑐superscriptsubscript𝑡1𝑘𝛼superscriptsubscript𝜔𝑡w(G^{*},c^{*})=\sum_{t=1}^{k}\min\{\alpha,\omega_{t}^{*}\}. (A.6)

All that remains is to show that (A.6) is equal to the expression in (A.1). In other words, we will obtain the expression for ωtsuperscriptsubscript𝜔𝑡\omega_{t}^{*}.

Recall that in the generation process of Gsuperscript𝐺G^{*}, any newcomer node xn+tsuperscript𝑥𝑛𝑡x^{n+t} connects to n1𝑛1n-1 helper nodes {xm}m=t+1n+t1superscriptsubscriptsuperscript𝑥𝑚𝑚𝑡1𝑛𝑡1\{x^{m}\}_{m=t+1}^{n+t-1} to regenerate xtsuperscript𝑥𝑡x^{t}. Among the n1𝑛1n-1 helper nodes, the nI1subscript𝑛𝐼1n_{I}-1 nodes reside in the same cluster with xtsuperscript𝑥𝑡x^{t}, while the nnI𝑛subscript𝑛𝐼n-n_{I} nodes are in other clusters. From our system setting in Section III-A, the helper nodes in the same cluster as the failed node help by βIsubscript𝛽𝐼\beta_{I}, while the helper nodes in other clusters help by βcsubscript𝛽𝑐\beta_{c}. Therefore, the total repair bandwidth to regenerate any failed node is

γ=(nI1)βI+(nnI)βc𝛾subscript𝑛𝐼1subscript𝛽𝐼𝑛subscript𝑛𝐼subscript𝛽𝑐\gamma=(n_{I}-1)\beta_{I}+(n-n_{I})\beta_{c} (A.7)

as in (2).

The newcomer node xinn+1subscriptsuperscript𝑥𝑛1𝑖𝑛x^{n+1}_{in} connects to {xoutm}m=2nsuperscriptsubscriptsuperscriptsubscript𝑥𝑜𝑢𝑡𝑚𝑚2𝑛\{x_{out}^{m}\}_{m=2}^{n}, all of which are included in U0subscript𝑈0U_{0}. Therefore, ω1=γ=(nI1)βI+(nnI)βcsuperscriptsubscript𝜔1𝛾subscript𝑛𝐼1subscript𝛽𝐼𝑛subscript𝑛𝐼subscript𝛽𝑐\omega_{1}^{*}=\gamma=(n_{I}-1)\beta_{I}+(n-n_{I})\beta_{c} holds. Next, xinn+2subscriptsuperscript𝑥𝑛2𝑖𝑛x^{n+2}_{in} connects to n2𝑛2n-2 nodes {xoutm}m=3nsuperscriptsubscriptsuperscriptsubscript𝑥𝑜𝑢𝑡𝑚𝑚3𝑛\{x_{out}^{m}\}_{m=3}^{n} from U0subscript𝑈0U_{0} and one node xoutn+1superscriptsubscript𝑥𝑜𝑢𝑡𝑛1x_{out}^{n+1} from U¯¯𝑈\mkern 1.5mu\overline{\mkern-1.5muU\mkern-1.5mu}\mkern 1.5mu. Define variable βlmsubscript𝛽𝑙𝑚\beta_{lm} as the repair bandwidth from xoutn+lsuperscriptsubscript𝑥𝑜𝑢𝑡𝑛𝑙x_{out}^{n+l} to xinn+msuperscriptsubscript𝑥𝑖𝑛𝑛𝑚x_{in}^{n+m}. Then, ω2=γβ12superscriptsubscript𝜔2𝛾subscript𝛽12\omega_{2}^{*}=\gamma-\beta_{12}. From (A.3), we have xn+1=N(1,1)superscript𝑥𝑛1𝑁11x^{n+1}=N(1,1) and xn+2=N(1,2)superscript𝑥𝑛2𝑁12x^{n+2}=N(1,2). Therefore, xn+1superscript𝑥𝑛1x^{n+1} and xn+2superscript𝑥𝑛2x^{n+2} are in different clusters, which result in β12=βcsubscript𝛽12subscript𝛽𝑐\beta_{12}=\beta_{c}. Therefore, ω2=γβ12=γβc.superscriptsubscript𝜔2𝛾subscript𝛽12𝛾subscript𝛽𝑐\omega_{2}^{*}=\gamma-\beta_{12}=\gamma-\beta_{c}.

Refer to caption
Figure 18: The cut-set c=(U,U¯)superscript𝑐𝑈¯𝑈c^{*}=(U,\mkern 1.5mu\overline{\mkern-1.5muU\mkern-1.5mu}\mkern 1.5mu) for the graph Gsuperscript𝐺G^{*}

In general, xinn+tsubscriptsuperscript𝑥𝑛𝑡𝑖𝑛x^{n+t}_{in} connects to nt𝑛𝑡n-t nodes {xoutm}m=t+1nsuperscriptsubscriptsuperscriptsubscript𝑥𝑜𝑢𝑡𝑚𝑚𝑡1𝑛\{x_{out}^{m}\}_{m=t+1}^{n} from U0subscript𝑈0U_{0}, and t1𝑡1t-1 nodes {xoutn+m}m=1t1superscriptsubscriptsuperscriptsubscript𝑥𝑜𝑢𝑡𝑛𝑚𝑚1𝑡1\{x_{out}^{n+m}\}_{m=1}^{t-1} from U¯¯𝑈\mkern 1.5mu\overline{\mkern-1.5muU\mkern-1.5mu}\mkern 1.5mu. Thus, ωtsuperscriptsubscript𝜔𝑡\omega_{t}^{*} for t[k]𝑡delimited-[]𝑘t\in[k] can be expressed as ωt=γl=1t1βltsuperscriptsubscript𝜔𝑡𝛾superscriptsubscript𝑙1𝑡1subscript𝛽𝑙𝑡\omega_{t}^{*}=\gamma-\sum_{l=1}^{t-1}\beta_{lt} where

βlt={βI, if xn+l and xn+t are in the same clusterβc, otherwise.subscript𝛽𝑙𝑡casessubscript𝛽𝐼 if superscript𝑥𝑛𝑙 and superscript𝑥𝑛𝑡 are in the same clustersubscript𝛽𝑐 otherwise.\beta_{lt}=\begin{cases}\beta_{I},&\text{ if }x^{n+l}\text{ and }x^{n+t}\text{ are in the same cluster}\\ \beta_{c},&\text{ otherwise.}\end{cases}

Recall Fig. 16. For arbitrary newcomer node xn+t=N(it,jt)superscript𝑥𝑛𝑡𝑁subscript𝑖𝑡subscript𝑗𝑡x^{n+t}=N(i_{t},j_{t}), the set {xn+m}m=1t1superscriptsubscriptsuperscript𝑥𝑛𝑚𝑚1𝑡1\{x^{n+m}\}_{m=1}^{t-1} contains it1subscript𝑖𝑡1i_{t}-1 nodes which reside in the same cluster with xn+tsuperscript𝑥𝑛𝑡x^{n+t}, and tit𝑡subscript𝑖𝑡t-i_{t} nodes in other clusters. Therefore, ωtsuperscriptsubscript𝜔𝑡\omega_{t}^{*} can be expressed as

ωt=γ(it1)βI(tit)βcsuperscriptsubscript𝜔𝑡𝛾subscript𝑖𝑡1subscript𝛽𝐼𝑡subscript𝑖𝑡subscript𝛽𝑐\omega_{t}^{*}=\gamma-(i_{t}-1)\beta_{I}-(t-i_{t})\beta_{c}

where itsubscript𝑖𝑡i_{t} is defined in (A.4). Combined with (A.7) and (A.5), we get

ωtsuperscriptsubscript𝜔𝑡\displaystyle\omega_{t}^{*} =(nIit)βI+(nnIt+it)βcabsentsubscript𝑛𝐼subscript𝑖𝑡subscript𝛽𝐼𝑛subscript𝑛𝐼𝑡subscript𝑖𝑡subscript𝛽𝑐\displaystyle=(n_{I}-i_{t})\beta_{I}+(n-n_{I}-t+i_{t})\beta_{c}
=(nIit)βI+(nnIjtm=1it1gm+it)βc.absentsubscript𝑛𝐼subscript𝑖𝑡subscript𝛽𝐼𝑛subscript𝑛𝐼subscript𝑗𝑡superscriptsubscript𝑚1subscript𝑖𝑡1subscript𝑔𝑚subscript𝑖𝑡subscript𝛽𝑐\displaystyle=(n_{I}-i_{t})\beta_{I}+(n-n_{I}-j_{t}-\sum_{m=1}^{i_{t}-1}g_{m}+i_{t})\beta_{c}.

Then, (A.6) can be expressed as

w(G,c)=𝑤superscript𝐺superscript𝑐absent\displaystyle w(G^{*},c^{*})=
i=1nItTimin{α,(nIi)βI+(nnIjtm=1i1gm+i)βc}superscriptsubscript𝑖1subscript𝑛𝐼subscript𝑡subscript𝑇𝑖𝛼subscript𝑛𝐼𝑖subscript𝛽𝐼𝑛subscript𝑛𝐼subscript𝑗𝑡superscriptsubscript𝑚1𝑖1subscript𝑔𝑚𝑖subscript𝛽𝑐\displaystyle\sum_{i=1}^{n_{I}}\sum_{t\in T_{i}}\min\{\alpha,(n_{I}-i)\beta_{I}+(n-n_{I}-j_{t}-\sum_{m=1}^{i-1}g_{m}+i)\beta_{c}\} (A.8)

where Ti={t[k]:it=i}subscript𝑇𝑖conditional-set𝑡delimited-[]𝑘subscript𝑖𝑡𝑖T_{i}=\{t\in[k]:i_{t}=i\}. From the definition of itsubscript𝑖𝑡i_{t} in (A.4), we have

Ti={m=1i1gm+1,m=1i1gm+2,,m=1igm}.subscript𝑇𝑖superscriptsubscript𝑚1𝑖1subscript𝑔𝑚1superscriptsubscript𝑚1𝑖1subscript𝑔𝑚2superscriptsubscript𝑚1𝑖subscript𝑔𝑚T_{i}=\{\sum_{m=1}^{i-1}g_{m}+1,\sum_{m=1}^{i-1}g_{m}+2,\cdots,\sum_{m=1}^{i}g_{m}\}.

Thus, jt=1,2,,gisubscript𝑗𝑡12subscript𝑔𝑖j_{t}=1,2,\cdots,g_{i} for tTi𝑡subscript𝑇𝑖t\in T_{i}. Therefore, (A) can be expressed as

w(G,c)=𝑤superscript𝐺superscript𝑐absent\displaystyle w(G^{*},c^{*})=
i=1nIj=1gimin{α,(nIi)βI+(nnIjm=1i1gm+i)βc},superscriptsubscript𝑖1subscript𝑛𝐼superscriptsubscript𝑗1subscript𝑔𝑖𝛼subscript𝑛𝐼𝑖subscript𝛽𝐼𝑛subscript𝑛𝐼𝑗superscriptsubscript𝑚1𝑖1subscript𝑔𝑚𝑖subscript𝛽𝑐\displaystyle\sum_{i=1}^{n_{I}}\sum_{j=1}^{g_{i}}\min\{\alpha,(n_{I}-i)\beta_{I}+(n-n_{I}-j-\sum_{m=1}^{i-1}g_{m}+i)\beta_{c}\},

which is identical to T𝑇T in (A.1), where ρisubscript𝜌𝑖\rho_{i} used in this equation is defined in (6). Therefore, the specified information flow graph Gsuperscript𝐺G^{*} and the specified cut-set csuperscript𝑐c^{*} satisfy ω(G,c)=t=1kmin{α,ωt}=T𝜔superscript𝐺superscript𝑐superscriptsubscript𝑡1𝑘𝛼superscriptsubscript𝜔𝑡𝑇\omega(G^{*},c^{*})=\sum_{t=1}^{k}\min\{\alpha,\omega_{t}^{*}\}=T.

Part II. Show that for every information flow graph G𝒢𝐺𝒢G\in\mathcal{G} and for every cut-set cC(G)𝑐𝐶𝐺c\in C(G), the cut-value w(G,c)𝑤𝐺𝑐w(G,c) is greater than or equal to T𝑇T in (A.1). In other words, G𝒢,cC(G)formulae-sequencefor-all𝐺𝒢for-all𝑐𝐶𝐺\forall G\in\mathcal{G},\forall c\in C(G), we have w(G,c)T𝑤𝐺𝑐𝑇w(G,c)\geq T.

The proof is divided into 2 sub-parts: Part II-1 and Part II-2.

Part II-1. Show that G𝒢,cC(G),formulae-sequencefor-all𝐺𝒢for-all𝑐𝐶𝐺\forall G\in\mathcal{G},\forall c\in C(G), we have w(G,c)B(G,c)𝑤𝐺𝑐𝐵𝐺𝑐w(G,c)\geq B(G,c) where B(G,c)𝐵𝐺𝑐B(G,c) is in (A.14):

Consider an arbitrary information flow graph G𝒢𝐺𝒢G\in\mathcal{G} and an arbitrary cut-set cC(G)𝑐𝐶𝐺c\in C(G) of the graph G𝐺G. Denote the cut-set as c=(U,U¯)𝑐𝑈¯𝑈c=(U,\mkern 1.5mu\overline{\mkern-1.5muU\mkern-1.5mu}\mkern 1.5mu). Consider an output node xoutisuperscriptsubscript𝑥𝑜𝑢𝑡𝑖x_{out}^{i} connected to DC𝐷𝐶DC. If xoutiUsuperscriptsubscript𝑥𝑜𝑢𝑡𝑖𝑈x_{out}^{i}\in U, then the cut-value w(G,c)𝑤𝐺𝑐w(G,c) is infinity, which is a trivial case for proving w(G,c)B(G,c)𝑤𝐺𝑐𝐵𝐺𝑐w(G,c)\geq B(G,c). Therefore, the k𝑘k output nodes connected to DC𝐷𝐶DC are assumed to be in U¯¯𝑈\mkern 1.5mu\overline{\mkern-1.5muU\mkern-1.5mu}\mkern 1.5mu. In other words, at least k𝑘k output nodes exist in U¯¯𝑈\mkern 1.5mu\overline{\mkern-1.5muU\mkern-1.5mu}\mkern 1.5mu. Note that every directed acyclic graph can be topologically sorted [34], where vertex u𝑢u is followed by vertex v𝑣v if there exists a directed edge from u𝑢u to v𝑣v. Consider labeling the topologically first k𝑘k output nodes in U¯¯𝑈\mkern 1.5mu\overline{\mkern-1.5muU\mkern-1.5mu}\mkern 1.5mu as vout1,,voutksuperscriptsubscript𝑣𝑜𝑢𝑡1superscriptsubscript𝑣𝑜𝑢𝑡𝑘v_{out}^{1},\cdots,v_{out}^{k}. Similar to the notation for a storage node xi=(xini,xouti)superscript𝑥𝑖subscriptsuperscript𝑥𝑖𝑖𝑛subscriptsuperscript𝑥𝑖𝑜𝑢𝑡x^{i}=(x^{i}_{in},x^{i}_{out}) in Section II-B, we denote the storage node which contains voutisubscriptsuperscript𝑣𝑖𝑜𝑢𝑡v^{i}_{out} as vi=(vini,vouti)superscript𝑣𝑖subscriptsuperscript𝑣𝑖𝑖𝑛subscriptsuperscript𝑣𝑖𝑜𝑢𝑡v^{i}=(v^{i}_{in},v^{i}_{out}). Then, the set of ordered klimit-from𝑘k-tuples (vi)i=1ksuperscriptsubscriptsuperscript𝑣𝑖𝑖1𝑘(v^{i})_{i=1}^{k} can be represented as

𝒱k={(v1,v2,,vk):\displaystyle\mathcal{V}_{k}=\{(v^{1},v^{2},\cdots,v^{k}): vt𝒩 for t[k]superscript𝑣𝑡𝒩 for 𝑡delimited-[]𝑘\displaystyle\quad v^{t}\in\mathcal{N}\text{ for }t\in[k]
vt1vt2 for t1t2}.\displaystyle\quad v^{t_{1}}\neq v^{t_{2}}\text{ for }t_{1}\neq t_{2}\}. (A.9)

We also define uisubscript𝑢𝑖u_{i}, the sum of capacities of edges from U𝑈U to vinisuperscriptsubscript𝑣𝑖𝑛𝑖v_{in}^{i}. See Fig. 19.

Refer to caption
Figure 19: Arbitrary information flow graph G𝒢𝐺𝒢G\in\mathcal{G} and arbitrary cut-set cC(G)𝑐𝐶𝐺c\in C(G)

If vin1Usuperscriptsubscript𝑣𝑖𝑛1𝑈v_{in}^{1}\in U, then the cut-set c=(U,U¯)𝑐𝑈¯𝑈c=(U,\mkern 1.5mu\overline{\mkern-1.5muU\mkern-1.5mu}\mkern 1.5mu) should include the edge from vin1superscriptsubscript𝑣𝑖𝑛1v_{in}^{1} to vout1superscriptsubscript𝑣𝑜𝑢𝑡1v_{out}^{1}, which has the edge capacity α𝛼\alpha. Otherwise (i.e., vin1U¯superscriptsubscript𝑣𝑖𝑛1¯𝑈v_{in}^{1}\in\mkern 1.5mu\overline{\mkern-1.5muU\mkern-1.5mu}\mkern 1.5mu), the cut-set c=(U,U¯)𝑐𝑈¯𝑈c=(U,\mkern 1.5mu\overline{\mkern-1.5muU\mkern-1.5mu}\mkern 1.5mu) should include the edges from U𝑈U to vin1superscriptsubscript𝑣𝑖𝑛1v_{in}^{1}. If vin1superscriptsubscript𝑣𝑖𝑛1v_{in}^{1} node is directly connected to the source node S𝑆S, the cut-value w(G,c)𝑤𝐺𝑐w(G,c) is infinity (trivial case for proving w(G,c)B(G,c)𝑤𝐺𝑐𝐵𝐺𝑐w(G,c)\geq B(G,c)). Therefore, vin1superscriptsubscript𝑣𝑖𝑛1v_{in}^{1} node is assumed to be a newcomer node helped by n1𝑛1n-1 helper nodes. Note that all helper nodes of vin1superscriptsubscript𝑣𝑖𝑛1v_{in}^{1} are in U𝑈U, since vout1superscriptsubscript𝑣𝑜𝑢𝑡1v_{out}^{1} is the topologically first output node in U¯¯𝑈\mkern 1.5mu\overline{\mkern-1.5muU\mkern-1.5mu}\mkern 1.5mu. Thus, the cut-set c𝑐c should include the edges from U𝑈U to vin1superscriptsubscript𝑣𝑖𝑛1v_{in}^{1}, where the sum of capacities of these edges are

u1=γ=(nI1)βI+(nnI)βc.subscript𝑢1𝛾subscript𝑛𝐼1subscript𝛽𝐼𝑛subscript𝑛𝐼subscript𝛽𝑐u_{1}=\gamma=(n_{I}-1)\beta_{I}+(n-n_{I})\beta_{c}.

If vin2Usuperscriptsubscript𝑣𝑖𝑛2𝑈v_{in}^{2}\in U, then the cut-set c=(U,U¯)𝑐𝑈¯𝑈c=(U,\mkern 1.5mu\overline{\mkern-1.5muU\mkern-1.5mu}\mkern 1.5mu) should include the edge from vin2superscriptsubscript𝑣𝑖𝑛2v_{in}^{2} to vout2superscriptsubscript𝑣𝑜𝑢𝑡2v_{out}^{2}, which has the edge capacity α𝛼\alpha. Otherwise (i.e., vin2U¯superscriptsubscript𝑣𝑖𝑛2¯𝑈v_{in}^{2}\in\mkern 1.5mu\overline{\mkern-1.5muU\mkern-1.5mu}\mkern 1.5mu), the cut-set c=(U,U¯)𝑐𝑈¯𝑈c=(U,\mkern 1.5mu\overline{\mkern-1.5muU\mkern-1.5mu}\mkern 1.5mu) should include the edges from U𝑈U to vin2superscriptsubscript𝑣𝑖𝑛2v_{in}^{2}. As we discussed in the case of vin1superscriptsubscript𝑣𝑖𝑛1v_{in}^{1}, we can assume vin2superscriptsubscript𝑣𝑖𝑛2v_{in}^{2} is a newcomer node helped by n1𝑛1n-1 helper nodes. Since vin2superscriptsubscript𝑣𝑖𝑛2v_{in}^{2} is the topologically second node in U¯¯𝑈\mkern 1.5mu\overline{\mkern-1.5muU\mkern-1.5mu}\mkern 1.5mu, it may have one helper node vout1U¯superscriptsubscript𝑣𝑜𝑢𝑡1¯𝑈v_{out}^{1}\in\mkern 1.5mu\overline{\mkern-1.5muU\mkern-1.5mu}\mkern 1.5mu; at least n2𝑛2n-2 helper nodes in U𝑈U help to generate vin2superscriptsubscript𝑣𝑖𝑛2v_{in}^{2}. Note that the total amount of data coming into vin2superscriptsubscript𝑣𝑖𝑛2v_{in}^{2} is γ=(nI1)βI+(nnI)βc𝛾subscript𝑛𝐼1subscript𝛽𝐼𝑛subscript𝑛𝐼subscript𝛽𝑐\gamma=(n_{I}-1)\beta_{I}+(n-n_{I})\beta_{c}, while the amount of information coming from vout1superscriptsubscript𝑣𝑜𝑢𝑡1v_{out}^{1} to vin2superscriptsubscript𝑣𝑖𝑛2v_{in}^{2}, denoted β12subscript𝛽12\beta_{12}, is as follows: if v1superscript𝑣1v^{1} and v2superscript𝑣2v^{2} are in the same cluster, β12=βIsubscript𝛽12subscript𝛽𝐼\beta_{12}=\beta_{I}, otherwise β12=βcsubscript𝛽12subscript𝛽𝑐\beta_{12}=\beta_{c}. Recall that the cut-set should include the edges from U𝑈U to vin2superscriptsubscript𝑣𝑖𝑛2v_{in}^{2}. The sum of capacities of these edges are

u2γβ12,subscript𝑢2𝛾subscript𝛽12u_{2}\geq\gamma-\beta_{12},

while the equality holds if and only if vout1superscriptsubscript𝑣𝑜𝑢𝑡1v_{out}^{1} helps vin2superscriptsubscript𝑣𝑖𝑛2v_{in}^{2}. In a similar way, for i[k]𝑖delimited-[]𝑘i\in[k], uisubscript𝑢𝑖u_{i} can be bounded as

uiωisubscript𝑢𝑖subscript𝜔𝑖u_{i}\geq\omega_{i} (A.10)

where

ωisubscript𝜔𝑖\displaystyle\omega_{i} =γj=1i1βji,absent𝛾superscriptsubscript𝑗1𝑖1subscript𝛽𝑗𝑖\displaystyle=\gamma-\displaystyle\sum_{j=1}^{i-1}\beta_{ji}, (A.11)
βjisubscript𝛽𝑗𝑖\displaystyle\beta_{ji} ={βI,vj and vi are in the same clusterβc,otherwise.absentcasessubscript𝛽𝐼superscript𝑣𝑗 and superscript𝑣𝑖 are in the same clustersubscript𝛽𝑐otherwise.\displaystyle=\begin{cases}\beta_{I},&v^{j}\text{ and }v^{i}\text{ are in the same cluster}\\ \beta_{c},&\text{otherwise.}\\ \end{cases} (A.12)

The equality in (A.10) holds if and only if voutjsuperscriptsubscript𝑣𝑜𝑢𝑡𝑗v_{out}^{j} helps vinisuperscriptsubscript𝑣𝑖𝑛𝑖v_{in}^{i} for j[i1]𝑗delimited-[]𝑖1j\in[i-1].

Thus, voutisuperscriptsubscript𝑣𝑜𝑢𝑡𝑖v_{out}^{i} contributes at least min{α,ωi}𝛼subscript𝜔𝑖\min\{\alpha,\omega_{i}\} to the cut value, for i[k]𝑖delimited-[]𝑘i\in[k]. In summary, for arbitrary graph G𝒢𝐺𝒢G\in\mathcal{G}, an arbitrary cut-set c𝑐c has cut-value w(G,c)𝑤𝐺𝑐w(G,c) of at least i=1kmin{α,ωi}superscriptsubscript𝑖1𝑘𝛼subscript𝜔𝑖\sum_{i=1}^{k}\min\{\alpha,\omega_{i}\}:

w(G,c)i=1kmin{α,ωi},G𝒢,cC(G).formulae-sequence𝑤𝐺𝑐superscriptsubscript𝑖1𝑘𝛼subscript𝜔𝑖formulae-sequencefor-all𝐺𝒢for-all𝑐𝐶𝐺w(G,c)\geq\sum_{i=1}^{k}\min\{\alpha,\omega_{i}\},\quad\quad\forall G\in\mathcal{G},\forall c\in C(G). (A.13)

Note that {ωi}subscript𝜔𝑖\{\omega_{i}\} depends on the relative position of {vouti}i=1ksuperscriptsubscriptsuperscriptsubscript𝑣𝑜𝑢𝑡𝑖𝑖1𝑘\{v_{out}^{i}\}_{i=1}^{k}, which is determined when an arbitrary information flow graph G𝒢𝐺𝒢G\in\mathcal{G} and arbitrary cut-set cC(G)𝑐𝐶𝐺c\in C(G) are specified. This relationship is illustrated in Fig. 20. Therefore, we define

B(G,c)=i=1kmin{α,ωi}𝐵𝐺𝑐superscriptsubscript𝑖1𝑘𝛼subscript𝜔𝑖B(G,c)=\sum_{i=1}^{k}\min\{\alpha,\omega_{i}\} (A.14)

for arbitrary G𝒢𝐺𝒢G\in\mathcal{G} and arbitrary cC(G)𝑐𝐶𝐺c\in C(G). Combining (A.13) and (A.14) completes the proof part II-1.

Refer to caption
Figure 20: Dependency graph for variables in the proof of Part II.

Part II-2. minG𝒢mincC(G)B(G,c)=Rsubscript𝐺𝒢subscript𝑐𝐶𝐺𝐵𝐺𝑐𝑅\displaystyle\min_{G\in\mathcal{G}}\ \min_{c\in C(G)}B(G,c)=R:

Assume that α𝛼\alpha and k𝑘k are fixed. See Fig. 20. Note that for a given graph G𝒢𝐺𝒢G\in\mathcal{G} and a cut-set c=(U,U¯)𝑐𝑈¯𝑈c=(U,\mkern 1.5mu\overline{\mkern-1.5muU\mkern-1.5mu}\mkern 1.5mu) with cC(G)𝑐𝐶𝐺c\in C(G), the sequence of topologically first k𝑘k output nodes (vouti)i=1ksuperscriptsubscriptsuperscriptsubscript𝑣𝑜𝑢𝑡𝑖𝑖1𝑘(v_{out}^{i})_{i=1}^{k} in U¯¯𝑈\mkern 1.5mu\overline{\mkern-1.5muU\mkern-1.5mu}\mkern 1.5mu is determined. Moreover, for a given sequence (vouti)i=1ksuperscriptsubscriptsuperscriptsubscript𝑣𝑜𝑢𝑡𝑖𝑖1𝑘(v_{out}^{i})_{i=1}^{k}, we have a fixed (ωi)i=1ksuperscriptsubscriptsubscript𝜔𝑖𝑖1𝑘(\omega_{i})_{i=1}^{k}, which determines B(G,c)𝐵𝐺𝑐B(G,c) in (A.14). Thus, minG𝒢mincC(G)B(G,c)subscript𝐺𝒢subscript𝑐𝐶𝐺𝐵𝐺𝑐\displaystyle\min_{G\in\mathcal{G}}\ \min_{c\in C(G)}B(G,c) can be obtained by finding the optimal (vouti)i=1ksuperscriptsubscriptsuperscriptsubscript𝑣𝑜𝑢𝑡𝑖𝑖1𝑘(v_{out}^{i})_{i=1}^{k} sequence which minimizes B(G,c)𝐵𝐺𝑐B(G,c). It is identical to finding the optimal (vi)i=1ksuperscriptsubscriptsuperscript𝑣𝑖𝑖1𝑘(v^{i})_{i=1}^{k}, the sequence of k𝑘k different nodes out of n𝑛n existing nodes in the system. Therefore, based on (A.14) and (A.11), we have

minG𝒢mincC(G)B(G,c)=min(vi)i=1k𝒱k(i=1kmin{α,γj=1i1βji})subscript𝐺𝒢subscript𝑐𝐶𝐺𝐵𝐺𝑐subscriptsuperscriptsubscriptsuperscript𝑣𝑖𝑖1𝑘subscript𝒱𝑘superscriptsubscript𝑖1𝑘𝛼𝛾superscriptsubscript𝑗1𝑖1subscript𝛽𝑗𝑖\displaystyle\min_{G\in\mathcal{G}}\ \min_{c\in C(G)}B(G,c)=\displaystyle\min_{(v^{i})_{i=1}^{k}\in\mathcal{V}_{k}}\left(\sum_{i=1}^{k}\min\{\alpha,\gamma-\sum_{j=1}^{i-1}\beta_{ji}\}\right) (A.15)

where

βji={βI,vj and vi are in the same clusterβc,otherwise.subscript𝛽𝑗𝑖casessubscript𝛽𝐼superscript𝑣𝑗 and superscript𝑣𝑖 are in the same clustersubscript𝛽𝑐otherwise.\beta_{ji}=\begin{cases}\beta_{I},&v^{j}\text{ and }v^{i}\text{ are in the same cluster}\\ \beta_{c},&\text{otherwise.}\\ \end{cases} (A.16)

holds as defined in (A.12), and 𝒱ksubscript𝒱𝑘\mathcal{V}_{k} is defined in (A). In order to obtain the solution for RHS of (A.15), all we need to do is to find the optimal sequence (vi)i=1ksuperscriptsubscriptsuperscript𝑣𝑖𝑖1𝑘(v^{i})_{i=1}^{k} of k𝑘k different nodes, which can be divided into two sub-problems: i)i) finding the optimal way of selecting k𝑘k nodes {vi}i=1ksuperscriptsubscriptsuperscript𝑣𝑖𝑖1𝑘\{v^{i}\}_{i=1}^{k} out of n𝑛n nodes, and ii)ii) finding the optimal order of selected k𝑘k nodes. Note that there are (nk)binomial𝑛𝑘n\choose k selection methods and k!𝑘k! ordering methods. Each selection method can be assigned to a selection vector 𝒔𝒔\bm{s} defined in Definition 1, and each ordering method can be assigned to an ordering vector 𝝅𝝅\bm{\pi} defined in Definition 2.

First, we define a selection vector 𝒔𝒔\bm{s} for a given {vi}i=1ksuperscriptsubscriptsuperscript𝑣𝑖𝑖1𝑘\{v^{i}\}_{i=1}^{k}.

Definition 1.

Assume arbitrary k𝑘k nodes are selected as {vi}i=1ksuperscriptsubscriptsuperscript𝑣𝑖𝑖1𝑘\{v^{i}\}_{i=1}^{k}. Label each cluster by the number of selected nodes in a descending order. In other words, the 1stsuperscript1𝑠𝑡1^{st} cluster contains a maximum number of selected nodes, and the Lthsuperscript𝐿𝑡L^{th} cluster contains a minimum number of selected nodes. Under this setting, define the selection vector 𝐬=[s1,s2,,sL]𝐬subscript𝑠1subscript𝑠2subscript𝑠𝐿\bm{s}=[s_{1},s_{2},\cdots,s_{L}] where sisubscript𝑠𝑖s_{i} is the number of selected nodes in the ithsuperscript𝑖𝑡i^{th} cluster.

Refer to caption
Figure 21: Obtaining the selection vector for given k𝑘k output nodes {vouti}i=1ksuperscriptsubscriptsuperscriptsubscript𝑣𝑜𝑢𝑡𝑖𝑖1𝑘\{v_{out}^{i}\}_{i=1}^{k} (n=15,k=8,L=3formulae-sequence𝑛15formulae-sequence𝑘8𝐿3n=15,k=8,L=3)

Fig. 21 shows an example of selection vector 𝒔𝒔\bm{s} corresponding to the selected k𝑘k nodes {vi}i=1ksuperscriptsubscriptsuperscript𝑣𝑖𝑖1𝑘\{v^{i}\}_{i=1}^{k}. From the definition of the selection vector, the set of possible selection vectors can be specified as follows.

𝒮={𝒔\displaystyle\mathcal{S}=\big{\{}\bm{s} =[s1,,sL]:0sinI,si+1si,i=1Lsi=k}.\displaystyle=[s_{1},\cdots,s_{L}]:0\leq s_{i}\leq n_{I},s_{i+1}\leq s_{i},\sum_{i=1}^{L}s_{i}=k\big{\}}.

Note that even though (nk)binomial𝑛𝑘{n\choose k} different selections exist, the {ωi}subscript𝜔𝑖\{\omega_{i}\} values in (A.11) are only determined by the corresponding selection vector 𝒔𝒔\bm{s}. This is because {ωi}subscript𝜔𝑖\{\omega_{i}\} depends only on the relative positions of {vi}i=1ksuperscriptsubscriptsuperscript𝑣𝑖𝑖1𝑘\{v^{i}\}_{i=1}^{k}, whether they are in the same cluster or in different clusters. Therefore, comparing the {ωi}subscript𝜔𝑖\{\omega_{i}\} values of all |𝒮|𝒮|\mathcal{S}| possible selection vectors 𝒔𝒔\bm{s} is enough; it is not necessary to compare the {ωi}subscript𝜔𝑖\{\omega_{i}\} values of (nk)binomial𝑛𝑘{n\choose k} selection methods. Now, we define the ordering vector 𝝅𝝅\bm{\pi} for a given selection vector 𝒔𝒔\bm{s}.

Definition 2.

Let the locations of k𝑘k nodes {vi}i=1ksuperscriptsubscriptsuperscript𝑣𝑖𝑖1𝑘\{v^{i}\}_{i=1}^{k} be fixed with a corresponding selection vector 𝐬=[s1,,sL]𝐬subscript𝑠1subscript𝑠𝐿\bm{s}=[s_{1},\cdots,s_{L}]. Then, for arbitrary ordering of the selected k𝑘k nodes, define the ordering vector 𝛑=[π1,,πk]𝛑subscript𝜋1subscript𝜋𝑘\bm{\pi}=[\pi_{1},\cdots,\pi_{k}] where πisubscript𝜋𝑖\pi_{i} is the index of the cluster which contains visuperscript𝑣𝑖v^{i}.

For a given 𝒔𝒔\bm{s}, the ordering vector 𝝅𝝅\bm{\pi} corresponding to an arbitrary ordering of k𝑘k nodes is illustrated in Fig. 22. In this figure (and the following figures in this paper), the number i𝑖i written inside each node means that the node is visuperscript𝑣𝑖v^{i}. From the definition, an ordering vector 𝝅𝝅\bm{\pi} has slsubscript𝑠𝑙s_{l} components with value l𝑙l, for all l[L]𝑙delimited-[]𝐿l\in[L]. The set of possible ordering vectors can be specified as

Π(𝒔)={𝝅=[π1,,πk]:i=1k𝟙πi=l=sl,l[L]}Π𝒔conditional-set𝝅subscript𝜋1subscript𝜋𝑘formulae-sequencesuperscriptsubscript𝑖1𝑘subscript1subscript𝜋𝑖𝑙subscript𝑠𝑙for-all𝑙delimited-[]𝐿\Pi(\bm{s})=\big{\{}\bm{\pi}=[\pi_{1},\cdots,\pi_{k}]:\sum_{i=1}^{k}\mathds{1}_{\pi_{i}=l}=s_{l},\ \forall l\in[L]\big{\}} (A.17)

Note that for given k𝑘k selected nodes, there exists k!𝑘k! different ordering methods. However, the {ωi}subscript𝜔𝑖\{\omega_{i}\} values in (A.11) are only determined by the corresponding ordering vector 𝝅Π(𝒔)𝝅Π𝒔\bm{\pi}\in\Pi(\bm{s}), by similar reasoning for compressing (nk)binomial𝑛𝑘{n\choose k} selection methods to |𝒮|𝒮|\mathcal{S}| selection vectors. Therefore, comparing the {ωi}subscript𝜔𝑖\{\omega_{i}\} values of all possible ordering vectors 𝝅𝝅\bm{\pi} is enough; it is not necessary to compare the {ωi}subscript𝜔𝑖\{\omega_{i}\} values of all k!𝑘k! ordering methods.

Refer to caption
Figure 22: Obtaining the ordering vector for given an arbitrary order of k𝑘k output nodes (n=15,k=8,L=3formulae-sequence𝑛15formulae-sequence𝑘8𝐿3n=15,k=8,L=3)

Thus, finding the optimal sequence (vi)i=1ksuperscriptsubscriptsuperscript𝑣𝑖𝑖1𝑘(v^{i})_{i=1}^{k} is identical to specifying the optimal (𝒔,𝝅𝒔𝝅\bm{s,\pi}) pair, which is obtained as follows. Recall that from the definition of 𝝅=[π1,π2,,πk]𝝅subscript𝜋1subscript𝜋2subscript𝜋𝑘\bm{\pi}=[\pi_{1},\pi_{2},\cdots,\pi_{k}] in Definition 2, πi=πjsubscript𝜋𝑖subscript𝜋𝑗\pi_{i}=\pi_{j} holds if and only if visuperscript𝑣𝑖v^{i} and vjsuperscript𝑣𝑗v^{j} are in the same cluster. Therefore, βjisubscript𝛽𝑗𝑖\beta_{ji} in (A.16) can be expressed by using 𝝅𝝅\bm{\pi} notation:

βji(𝝅)={βI if πi=πjβc otherwisesubscript𝛽𝑗𝑖𝝅casessubscript𝛽𝐼 if subscript𝜋𝑖subscript𝜋𝑗subscript𝛽𝑐 otherwise\beta_{ji}(\bm{\pi})=\begin{cases}\beta_{I}&\text{ if }\pi_{i}=\pi_{j}\\ \beta_{c}&\text{ otherwise}\end{cases} (A.18)

Thus, the j=1i1βjisuperscriptsubscript𝑗1𝑖1subscript𝛽𝑗𝑖\sum_{j=1}^{i-1}\beta_{ji} term in (A.15) can be expressed as

j=1i1βji(𝝅)superscriptsubscript𝑗1𝑖1subscript𝛽𝑗𝑖𝝅\displaystyle\sum_{j=1}^{i-1}\beta_{ji}(\bm{\pi}) =(j=1i1𝟙πj=πi)βI+(i1j=1i1𝟙πj=πi)βc.absentsuperscriptsubscript𝑗1𝑖1subscript1subscript𝜋𝑗subscript𝜋𝑖subscript𝛽𝐼𝑖1superscriptsubscript𝑗1𝑖1subscript1subscript𝜋𝑗subscript𝜋𝑖subscript𝛽𝑐\displaystyle=(\sum_{j=1}^{i-1}\mathds{1}_{\pi_{j}=\pi_{i}})\beta_{I}+(i-1-\sum_{j=1}^{i-1}\mathds{1}_{\pi_{j}=\pi_{i}})\beta_{c}. (A.19)

Combining (2), (A.15) and (A.19), we have

minG𝒢mincC(G)B(G,c)=min𝒔Smin𝝅Π(𝒔)L(𝒔,𝝅)subscript𝐺𝒢subscript𝑐𝐶𝐺𝐵𝐺𝑐subscript𝒔𝑆subscript𝝅Π𝒔𝐿𝒔𝝅\displaystyle\min_{G\in\mathcal{G}}\ \min_{c\in C(G)}B(G,c)=\displaystyle\min_{\bm{s}\in S}\ \min_{\bm{\pi}\in\Pi(\bm{s})}L(\bm{s},\bm{\pi})

where

L(𝒔,𝝅)𝐿𝒔𝝅\displaystyle L(\bm{s},\bm{\pi}) =i=1kmin{α,ωi(𝝅)},absentsuperscriptsubscript𝑖1𝑘𝛼subscript𝜔𝑖𝝅\displaystyle=\sum_{i=1}^{k}\min\{\alpha,\omega_{i}(\bm{\pi})\}, (A.20)
ωi(𝝅)subscript𝜔𝑖𝝅\displaystyle\omega_{i}(\bm{\pi}) =γj=1i1βji(𝝅)=ai(𝝅)βI+(niai(𝝅))βc,absent𝛾superscriptsubscript𝑗1𝑖1subscript𝛽𝑗𝑖𝝅subscript𝑎𝑖𝝅subscript𝛽𝐼𝑛𝑖subscript𝑎𝑖𝝅subscript𝛽𝑐\displaystyle=\gamma-\sum_{j=1}^{i-1}\beta_{ji}(\bm{\pi})=a_{i}(\bm{\pi})\beta_{I}+(n-i-a_{i}(\bm{\pi}))\beta_{c}, (A.21)
ai(𝝅)subscript𝑎𝑖𝝅\displaystyle a_{i}(\bm{\pi}) =nI1j=1i1𝟙πj=πi.absentsubscript𝑛𝐼1superscriptsubscript𝑗1𝑖1subscript1subscript𝜋𝑗subscript𝜋𝑖\displaystyle=n_{I}-1-\sum_{j=1}^{i-1}\mathds{1}_{\pi_{j}=\pi_{i}}. (A.22)

Therefore, the rest of the proof in part II-2 shows that

min𝒔Smin𝝅Π(𝒔)L(𝒔,𝝅)=Rsubscript𝒔𝑆subscript𝝅Π𝒔𝐿𝒔𝝅𝑅\displaystyle\min_{\bm{s}\in S}\ \min_{\bm{\pi}\in\Pi(\bm{s})}L(\bm{s},\bm{\pi})=R (A.23)

holds. We begin by stating a property of ωi(𝝅)subscript𝜔𝑖𝝅\omega_{i}(\bm{\pi}) seen in (A.21).

Proposition 3.

Consider a fixed selection vector 𝐬𝐬\bm{s}. We claim that i=1kωi(𝛑)superscriptsubscript𝑖1𝑘subscript𝜔𝑖𝛑\sum_{i=1}^{k}\omega_{i}(\bm{\pi}) is constant irrespective of the ordering vector 𝛑Π(𝐬)𝛑Π𝐬\bm{\pi}\in\Pi(\bm{s}).

Proof.

Let 𝒔=[s1,,sL]𝒔subscript𝑠1subscript𝑠𝐿\bm{s}=[s_{1},\cdots,s_{L}]. For an arbitrary ordering vector 𝝅Π(𝒔)𝝅Π𝒔\bm{\pi}\in\Pi(\bm{s}), let bi(𝝅)=niai(𝝅)subscript𝑏𝑖𝝅𝑛𝑖subscript𝑎𝑖𝝅b_{i}(\bm{\pi})=n-i-a_{i}(\bm{\pi}) where ai(𝝅)subscript𝑎𝑖𝝅a_{i}(\bm{\pi}) is as given in (A.22). For simplicity, we denote ai(𝝅)subscript𝑎𝑖𝝅a_{i}(\bm{\pi}), bi(𝝅)subscript𝑏𝑖𝝅b_{i}(\bm{\pi}) and ωi(𝝅)subscript𝜔𝑖𝝅\omega_{i}(\bm{\pi}) as aisubscript𝑎𝑖a_{i}, bisubscript𝑏𝑖b_{i} and ωisubscript𝜔𝑖\omega_{i}, respectively. Then,

i=1k(ai+bi)=i=1k(ni)=constant (const.)superscriptsubscript𝑖1𝑘subscript𝑎𝑖subscript𝑏𝑖superscriptsubscript𝑖1𝑘𝑛𝑖constant (const.)\sum_{i=1}^{k}(a_{i}+b_{i})=\sum_{i=1}^{k}(n-i)=\textit{constant (const.)} (A.24)

for fixed n,k𝑛𝑘n,k. Note that

i=1kai=k(nI1)i=1kj=1i1𝟙πj=πisuperscriptsubscript𝑖1𝑘subscript𝑎𝑖𝑘subscript𝑛𝐼1superscriptsubscript𝑖1𝑘superscriptsubscript𝑗1𝑖1subscript1subscript𝜋𝑗subscript𝜋𝑖\sum_{i=1}^{k}a_{i}=k(n_{I}-1)-\sum_{i=1}^{k}\sum_{j=1}^{i-1}\mathds{1}_{\pi_{j}=\pi_{i}} (A.25)

from (A.22). Also, from the definition of Π(𝒔)Π𝒔\Pi(\bm{s}) in (A.17), an arbitrary ordering vector 𝝅Π(𝒔)𝝅Π𝒔\bm{\pi}\in\Pi(\bm{s}) has slsubscript𝑠𝑙s_{l} components with value l𝑙l, for all l[L]𝑙delimited-[]𝐿l\in[L]. If we define

Il(𝝅)={i[k]:πi=l},subscript𝐼𝑙𝝅conditional-set𝑖delimited-[]𝑘subscript𝜋𝑖𝑙I_{l}(\bm{\pi})=\{i\in[k]:\pi_{i}=l\}, (A.26)

then |Il(𝝅)|=slsubscript𝐼𝑙𝝅subscript𝑠𝑙|I_{l}(\bm{\pi})|=s_{l} holds for l[L]𝑙delimited-[]𝐿l\in[L]. Then,

iIl(𝝅)j=1i1𝟙πj=πi=0+1++(sl1)=t=0sl1t.subscript𝑖subscript𝐼𝑙𝝅superscriptsubscript𝑗1𝑖1subscript1subscript𝜋𝑗subscript𝜋𝑖01subscript𝑠𝑙1superscriptsubscript𝑡0subscript𝑠𝑙1𝑡\sum_{i\in I_{l}(\bm{\pi})}\sum_{j=1}^{i-1}\mathds{1}_{\pi_{j}=\pi_{i}}=0+1+\cdots+(s_{l}-1)=\sum_{t=0}^{s_{l}-1}t.

Therefore,

i=1kj=1i1𝟙πj=πi=l=1LiIl(𝝅)j=1i1𝟙πj=πi=l=1Lt=0sl1t=const.superscriptsubscript𝑖1𝑘superscriptsubscript𝑗1𝑖1subscript1subscript𝜋𝑗subscript𝜋𝑖superscriptsubscript𝑙1𝐿subscript𝑖subscript𝐼𝑙𝝅superscriptsubscript𝑗1𝑖1subscript1subscript𝜋𝑗subscript𝜋𝑖superscriptsubscript𝑙1𝐿superscriptsubscript𝑡0subscript𝑠𝑙1𝑡const.\sum_{i=1}^{k}\sum_{j=1}^{i-1}\mathds{1}_{\pi_{j}=\pi_{i}}=\sum_{l=1}^{L}\sum_{i\in I_{l}(\bm{\pi})}\sum_{j=1}^{i-1}\mathds{1}_{\pi_{j}=\pi_{i}}=\sum_{l=1}^{L}\sum_{t=0}^{s_{l}-1}t=\textit{const.}

for fixed L,𝒔𝐿𝒔L,\bm{s}. Combining with (A.25),

i=1kai=k(nI1)l=1Lt=0sl1t=const.superscriptsubscript𝑖1𝑘subscript𝑎𝑖𝑘subscript𝑛𝐼1superscriptsubscript𝑙1𝐿superscriptsubscript𝑡0subscript𝑠𝑙1𝑡const.\sum_{i=1}^{k}a_{i}=k(n_{I}-1)-\sum_{l=1}^{L}\sum_{t=0}^{s_{l}-1}t=\textit{const.} (A.27)

for fixed n,k,L,𝒔𝑛𝑘𝐿𝒔n,k,L,\bm{s}. From (A.24) and (A.27) we get

i=1kbi=const.superscriptsubscript𝑖1𝑘subscript𝑏𝑖const.\sum_{i=1}^{k}b_{i}=\textit{const.}

Therefore, from (A.21),

i=1kωi=(i=1kai)βI+(i=1kbi)βc=const.superscriptsubscript𝑖1𝑘subscript𝜔𝑖superscriptsubscript𝑖1𝑘subscript𝑎𝑖subscript𝛽𝐼superscriptsubscript𝑖1𝑘subscript𝑏𝑖subscript𝛽𝑐const.\sum_{i=1}^{k}\omega_{i}=(\sum_{i=1}^{k}a_{i})\beta_{I}+(\sum_{i=1}^{k}b_{i})\beta_{c}=\textit{const.}

for every ordering vector 𝝅Π(𝒔)𝝅Π𝒔\bm{\pi}\in\Pi(\bm{s}), if n,k,L,βI,βc𝑛𝑘𝐿subscript𝛽𝐼subscript𝛽𝑐n,k,L,\beta_{I},\beta_{c} and 𝒔𝒔\bm{s} are fixed. ∎

Now, we define a special ordering vector 𝝅vsubscript𝝅𝑣\bm{\pi}_{v} called vertical ordering vector, which is shown to be the optimal ordering vector 𝝅𝝅\bm{\pi} which minimizes L(𝒔,𝝅)𝐿𝒔𝝅L(\bm{s},\bm{\pi}) for an arbitrary selection vector 𝒔𝒔\bm{s}.

Definition 3.

For a given selection vector 𝐬𝒮𝐬𝒮\bm{s}\in\mathcal{S}, the corresponding vertical ordering vector 𝛑v(𝐬)subscript𝛑𝑣𝐬\bm{\pi}_{v}(\bm{s}), or simply denoted as 𝛑vsubscript𝛑𝑣\bm{\pi}_{v}, is defined as the output of Algorithm 1.

The vertical ordering vector 𝝅vsubscript𝝅𝑣\bm{\pi}_{v} is illustrated in Fig. 23, for a given selection vector 𝒔𝒔\bm{s} as an example. For 𝒔=[4,3,1]𝒔431\bm{s}=[4,3,1], Algorithm 1 produces the corresponding vertical ordering vector 𝝅v=[1,2,3,1,2,1,2,1]subscript𝝅𝑣12312121\bm{\pi}_{v}=[1,2,3,1,2,1,2,1]. Note that the order of k=8𝑘8k=8 output nodes is illustrated in Fig. 23, as the numbers inside each node. Although the vertical ordering vector 𝝅vsubscript𝝅𝑣\bm{\pi}_{v} depends on the selection vector 𝒔𝒔\bm{s}, we use simplified notation 𝝅vsubscript𝝅𝑣\bm{\pi}_{v} instead of 𝝅v(𝒔)subscript𝝅𝑣𝒔\bm{\pi}_{v}(\bm{s}). From Fig. 23, obtaining 𝝅vsubscript𝝅𝑣\bm{\pi}_{v} using Algorithm 1 can be analyzed as follows. Moving from the leftmost column to the rightmost column, the algorithm selects one node per cluster. After selecting all k𝑘k nodes, πisubscript𝜋𝑖\pi_{i} stores the index of the cluster which contains the ithsuperscript𝑖𝑡i^{th} selected node. Now, the following Lemma shows that the vertical ordering vector 𝝅vsubscript𝝅𝑣\bm{\pi}_{v} is optimal in the sense of minimizing L(𝒔,𝝅)𝐿𝒔𝝅L(\bm{s},\bm{\pi}) for an arbitrary selection vector 𝒔𝒔\bm{s}.

Algorithm 1 Generate vertical ordering 𝝅vsubscript𝝅𝑣\bm{\pi}_{v}
  Input: 𝒔=[s1,,sL]𝒔subscript𝑠1subscript𝑠𝐿\bm{s}=[s_{1},\cdots,s_{L}]
  Output: 𝝅v=[π1,,πk]subscript𝝅𝑣subscript𝜋1subscript𝜋𝑘\bm{\pi}_{v}=[\pi_{1},\cdots,\pi_{k}]
  Initialization: l1𝑙1l\leftarrow 1
  for i=1𝑖1i=1 to k𝑘k do
     if sl=0subscript𝑠𝑙0s_{l}=0 then
        l1𝑙1l\leftarrow 1
     end if
     πilsubscript𝜋𝑖𝑙\pi_{i}\leftarrow l       (Store the index of cluster)
     sπisπi1subscript𝑠subscript𝜋𝑖subscript𝑠subscript𝜋𝑖1s_{\pi_{i}}\leftarrow s_{\pi_{i}}-1    (Update the remaining node info.)
     l(lmodL)+1𝑙𝑙mod𝐿1l\leftarrow(l\ \mathrm{mod}\ L)+1    (Go to the next cluster)
  end for
Refer to caption
Figure 23: The vertical ordering vector 𝝅vsubscript𝝅𝑣\bm{\pi}_{v} for the given selection vector 𝒔=[4,3,1]𝒔431\bm{s}=[4,3,1] (for n=15,k=8,L=3formulae-sequence𝑛15formulae-sequence𝑘8𝐿3n=15,k=8,L=3 case)
Lemma 3.

Let 𝐬𝒮𝐬𝒮\bm{s}\in\mathcal{S} be an arbitrary selection vector. Then, a vertical ordering vector 𝛑vsubscript𝛑𝑣\bm{\pi}_{v} minimizes L(𝐬,𝛑)𝐿𝐬𝛑L(\bm{s},\bm{\pi}). In other words, L(𝐬,𝛑v)L(𝐬,𝛑)𝐿𝐬subscript𝛑𝑣𝐿𝐬𝛑L(\bm{s},\bm{\pi}_{v})\leq L(\bm{s},\bm{\pi}) holds for arbitrary 𝛑Π(𝐬)𝛑Π𝐬\bm{\pi}\in\Pi(\bm{s}).

Proof.

In the case of βc=0subscript𝛽𝑐0\beta_{c}=0, we show that L(𝒔,𝝅)𝐿𝒔𝝅L(\bm{s},\bm{\pi}) is constant for every 𝝅Π(𝒔)𝝅Π𝒔\bm{\pi}\in\Pi(\bm{s}). From (A.20),

L(𝒔,𝝅)=i=1kmin{α,ai(𝝅)βI}𝐿𝒔𝝅superscriptsubscript𝑖1𝑘𝛼subscript𝑎𝑖𝝅subscript𝛽𝐼L(\bm{s},\bm{\pi})=\sum_{i=1}^{k}\min\{\alpha,a_{i}(\bm{\pi})\beta_{I}\} (A.28)

holds for βc=0subscript𝛽𝑐0\beta_{c}=0. Using Il(𝝅)subscript𝐼𝑙𝝅I_{l}(\bm{\pi}) in (A.26), note that [k]delimited-[]𝑘[k] can be partitioned into L𝐿L disjoint subsets as [k]=l=1LIl(𝝅).delimited-[]𝑘superscriptsubscript𝑙1𝐿subscript𝐼𝑙𝝅[k]=\bigcup_{l=1}^{L}I_{l}(\bm{\pi}). Therefore, (A.28) can be written as

L(𝒔,𝝅)=l=1LiIl(𝝅)min{α,ai(𝝅)βI}.𝐿𝒔𝝅superscriptsubscript𝑙1𝐿subscript𝑖subscript𝐼𝑙𝝅𝛼subscript𝑎𝑖𝝅subscript𝛽𝐼L(\bm{s},\bm{\pi})=\sum_{l=1}^{L}\sum_{i\in I_{l}(\bm{\pi})}\min\{\alpha,a_{i}(\bm{\pi})\beta_{I}\}.

Recall that 𝝅Π(𝒔)𝝅Π𝒔\bm{\pi}\in\Pi(\bm{s}) contains slsubscript𝑠𝑙s_{l} components with value l𝑙l for l[L]𝑙delimited-[]𝐿l\in[L]. Thus, from (A.22),

iIl(𝝅){ai(𝝅)}={nI1,nI2,,nIsl}subscript𝑖subscript𝐼𝑙𝝅subscript𝑎𝑖𝝅subscript𝑛𝐼1subscript𝑛𝐼2subscript𝑛𝐼subscript𝑠𝑙\bigcup_{i\in I_{l}(\bm{\pi})}\{a_{i}(\bm{\pi})\}=\{n_{I}-1,n_{I}-2,\cdots,n_{I}-s_{l}\}

for l[L]𝑙delimited-[]𝐿l\in[L]. Therefore, iIl(𝝅)min{α,ai(𝝅)βI}subscript𝑖subscript𝐼𝑙𝝅𝛼subscript𝑎𝑖𝝅subscript𝛽𝐼\sum_{i\in I_{l}(\bm{\pi})}\min\{\alpha,a_{i}(\bm{\pi})\beta_{I}\} is constant 𝝅Π(𝒔)for-all𝝅Π𝒔\forall\bm{\pi}\in\Pi(\bm{s}) for arbitrary l[L]𝑙delimited-[]𝐿l\in[L]. In conclusion, L(𝒔,𝝅)𝐿𝒔𝝅L(\bm{s},\bm{\pi}) in (A.28) is constant irrespective of 𝝅Π(𝒔)𝝅Π𝒔\bm{\pi}\in\Pi(\bm{s}) for βc=0subscript𝛽𝑐0\beta_{c}=0.

The rest of the proof deals with the βc0subscript𝛽𝑐0\beta_{c}\neq 0 case. For a given arbitrary 𝒔𝒮𝒔𝒮\bm{s}\in\mathcal{S}, define two subsets of Π(𝒔)Π𝒔\Pi(\bm{s}) as

Πr={𝝅Π(𝒔):St(𝝅)St(𝝅),t[k],𝝅Π(𝒔)}subscriptΠ𝑟conditional-setsuperscript𝝅Π𝒔formulae-sequencesubscript𝑆𝑡𝝅subscript𝑆𝑡superscript𝝅formulae-sequencefor-all𝑡delimited-[]𝑘for-all𝝅Π𝒔\displaystyle\Pi_{r}=\{\bm{\pi^{*}}\in\Pi(\bm{s}):\ S_{t}(\bm{\pi})\leq S_{t}(\bm{\pi^{*}}),\forall t\in[k],\ \forall\bm{\pi}\in\Pi(\bm{s})\} (A.29)
Πm={𝝅Π(𝒔):L(𝒔,𝝅)L(𝒔,𝝅),𝝅Π(𝒔)}subscriptΠ𝑚conditional-setsuperscript𝝅Π𝒔formulae-sequence𝐿𝒔𝝅𝐿𝒔superscript𝝅for-all𝝅Π𝒔\displaystyle\Pi_{m}=\{\bm{\pi^{*}}\in\Pi(\bm{s}):L(\bm{s},\bm{\pi})\geq L(\bm{s},\bm{\pi}^{*}),\forall\bm{\pi}\in\Pi(\bm{s})\} (A.30)

where St(𝝅)=i=1twi(𝝅)subscript𝑆𝑡𝝅superscriptsubscript𝑖1𝑡subscript𝑤𝑖𝝅S_{t}(\bm{\pi})=\sum_{i=1}^{t}w_{i}(\bm{\pi}) is the running sum of wi(𝝅)subscript𝑤𝑖𝝅w_{i}(\bm{\pi}). Here, we call ΠrsubscriptΠ𝑟\Pi_{r} the running sum maximizer and ΠmsubscriptΠ𝑚\Pi_{m} the min-cut minimizer. Now the proof proceeds in two steps. The first step proves that the running sum maximizer minimizes min-cut, i.e., ΠrΠmsubscriptΠ𝑟subscriptΠ𝑚\Pi_{r}\subseteq\Pi_{m}. The second step proves that the vertical ordering vector is a running sum maximizer, i.e., 𝝅vΠrsubscript𝝅𝑣subscriptΠ𝑟\bm{\pi}_{v}\in\Pi_{r}.

Step 1. Prove ΠrΠmsubscriptΠ𝑟subscriptΠ𝑚\Pi_{r}\subseteq\Pi_{m}:

Define two index sets for a given ordering vector 𝝅𝝅\bm{\pi}:

ΩL(𝝅)subscriptΩ𝐿𝝅\displaystyle\Omega_{L}(\bm{\pi}) ={i[k]:wi(𝝅)α}absentconditional-set𝑖delimited-[]𝑘subscript𝑤𝑖𝝅𝛼\displaystyle=\{i\in[k]:w_{i}(\bm{\pi})\geq\alpha\}
Ωs(𝝅)subscriptΩ𝑠𝝅\displaystyle\Omega_{s}(\bm{\pi}) ={i[k]:wi(𝝅)<α}absentconditional-set𝑖delimited-[]𝑘subscript𝑤𝑖𝝅𝛼\displaystyle=\{i\in[k]:w_{i}(\bm{\pi})<\alpha\} (A.31)

Now define a set of ordering vectors as

Πp={𝝅Π(𝒔):ijiΩL(𝝅),jΩs(𝝅)}.subscriptΠ𝑝conditional-set𝝅Π𝒔formulae-sequence𝑖𝑗for-all𝑖subscriptΩ𝐿𝝅for-all𝑗subscriptΩ𝑠𝝅\Pi_{p}=\{\bm{\pi}\in\Pi(\bm{s}):i\leq j\ \forall i\in\Omega_{L}(\bm{\pi}),\ \forall j\in\Omega_{s}(\bm{\pi})\}. (A.32)

The rest of the proof is divided into 2 sub-steps.

Step 1-1. Prove ΠmΠpsubscriptΠ𝑚subscriptΠ𝑝\Pi_{m}\subseteq\Pi_{p} and ΠrΠpsubscriptΠ𝑟subscriptΠ𝑝\Pi_{r}\subseteq\Pi_{p} by transposition:

Consider arbitrary 𝝅=[π1,,πk]Πpc𝝅subscript𝜋1subscript𝜋𝑘superscriptsubscriptΠ𝑝𝑐\bm{\pi}=[\pi_{1},\cdots,\pi_{k}]\in\Pi_{p}^{c}. Use a short-hand notation ωisubscript𝜔𝑖\omega_{i} to represent ωi(𝝅)subscript𝜔𝑖𝝅\omega_{i}(\bm{\pi}) for i[k]𝑖delimited-[]𝑘i\in[k]. From (A.32), there exists i>j𝑖𝑗i>j such that iΩL(𝝅)𝑖subscriptΩ𝐿𝝅i\in\Omega_{L}(\bm{\pi}) and jΩs(𝝅)𝑗subscriptΩ𝑠𝝅j\in\Omega_{s}(\bm{\pi}). Therefore, there exists t[k1]𝑡delimited-[]𝑘1t\in[k-1] such that t+1ΩL(𝝅)𝑡1subscriptΩ𝐿𝝅t+1\in\Omega_{L}(\bm{\pi}) and tΩs(𝝅)𝑡subscriptΩ𝑠𝝅t\in\Omega_{s}(\bm{\pi}) hold. By (A),

ωt+1α>ωtsubscript𝜔𝑡1𝛼subscript𝜔𝑡\omega_{t+1}\geq\alpha>\omega_{t} (A.33)

for some t[k1]𝑡delimited-[]𝑘1t\in[k-1]. Note that from (A.21), πt=πt+1subscript𝜋𝑡subscript𝜋𝑡1\pi_{t}=\pi_{t+1} implies ωt+1=ωtβI<ωtsubscript𝜔𝑡1subscript𝜔𝑡subscript𝛽𝐼subscript𝜔𝑡\omega_{t+1}=\omega_{t}-\beta_{I}<\omega_{t}. Therefore,

πtπt+1subscript𝜋𝑡subscript𝜋𝑡1\pi_{t}\neq\pi_{t+1} (A.34)

should hold to satisfy (A.33). Define an ordering vector 𝝅=[π1,,πk]superscript𝝅bold-′subscriptsuperscript𝜋1subscriptsuperscript𝜋𝑘\bm{\pi^{\prime}}=[\pi^{\prime}_{1},\cdots,\pi^{\prime}_{k}] as

{πi=πiit,t+1πt=πt+1πt+1=πt.casessubscriptsuperscript𝜋𝑖subscript𝜋𝑖𝑖𝑡𝑡1subscriptsuperscript𝜋𝑡subscript𝜋𝑡1otherwisesubscriptsuperscript𝜋𝑡1subscript𝜋𝑡otherwise\displaystyle\begin{cases}\pi^{\prime}_{i}=\pi_{i}&i\neq t,t+1\\ \pi^{\prime}_{t}=\pi_{t+1}\\ \pi^{\prime}_{t+1}=\pi_{t}.\end{cases} (A.35)

Use a short-hand notation ωisubscriptsuperscript𝜔𝑖\omega^{\prime}_{i} to represent ωi(𝝅)subscript𝜔𝑖superscript𝝅\omega_{i}(\bm{\pi}^{\prime}) for i[k]𝑖delimited-[]𝑘i\in[k]. Note that {ωi}subscriptsuperscript𝜔𝑖\{\omega^{\prime}_{i}\} satisfies

{ωi=ωiit,t+1ωt=ωt+1+βcωt+1=ωtβccasessubscriptsuperscript𝜔𝑖subscript𝜔𝑖𝑖𝑡𝑡1subscriptsuperscript𝜔𝑡subscript𝜔𝑡1subscript𝛽𝑐otherwisesubscriptsuperscript𝜔𝑡1subscript𝜔𝑡subscript𝛽𝑐otherwise\displaystyle\begin{cases}\omega^{\prime}_{i}=\omega_{i}&i\neq t,t+1\\ \omega^{\prime}_{t}=\omega_{t+1}+\beta_{c}\\ \omega^{\prime}_{t+1}=\omega_{t}-\beta_{c}\end{cases} (A.36)

for the following reason. First, use simplified notations aisubscript𝑎𝑖a_{i} and aisubscriptsuperscript𝑎𝑖a^{\prime}_{i} to mean ai(𝝅)subscript𝑎𝑖𝝅a_{i}(\bm{\pi}) and ai(𝝅)subscript𝑎𝑖superscript𝝅a_{i}(\bm{\pi}^{\prime}), respectively. Then, using (A.22), (A.34) and (A.35), we have

atsuperscriptsubscript𝑎𝑡\displaystyle a_{t}^{\prime} =nI1j=1t1𝟙πj=πt=nI1j=1t1𝟙πj=πt+1absentsubscript𝑛𝐼1superscriptsubscript𝑗1𝑡1subscript1superscriptsubscript𝜋𝑗superscriptsubscript𝜋𝑡subscript𝑛𝐼1superscriptsubscript𝑗1𝑡1subscript1subscript𝜋𝑗subscript𝜋𝑡1\displaystyle=n_{I}-1-\sum_{j=1}^{t-1}\mathds{1}_{\pi_{j}^{\prime}=\pi_{t}^{\prime}}=n_{I}-1-\sum_{j=1}^{t-1}\mathds{1}_{\pi_{j}=\pi_{t+1}}
=nI1j=1t𝟙πj=πt+1=at+1.absentsubscript𝑛𝐼1superscriptsubscript𝑗1𝑡subscript1subscript𝜋𝑗subscript𝜋𝑡1subscript𝑎𝑡1\displaystyle=n_{I}-1-\sum_{j=1}^{t}\mathds{1}_{\pi_{j}=\pi_{t+1}}=a_{t+1}. (A.37)

Similarly,

at+1superscriptsubscript𝑎𝑡1\displaystyle a_{t+1}^{\prime} =nI1j=1t𝟙πj=πt+1=nI1j=1t1𝟙πj=πt+1absentsubscript𝑛𝐼1superscriptsubscript𝑗1𝑡subscript1superscriptsubscript𝜋𝑗superscriptsubscript𝜋𝑡1subscript𝑛𝐼1superscriptsubscript𝑗1𝑡1subscript1superscriptsubscript𝜋𝑗superscriptsubscript𝜋𝑡1\displaystyle=n_{I}-1-\sum_{j=1}^{t}\mathds{1}_{\pi_{j}^{\prime}=\pi_{t+1}^{\prime}}=n_{I}-1-\sum_{j=1}^{t-1}\mathds{1}_{\pi_{j}^{\prime}=\pi_{t+1}^{\prime}}
=nI1j=1t1𝟙πj=πt=at.absentsubscript𝑛𝐼1superscriptsubscript𝑗1𝑡1subscript1subscript𝜋𝑗subscript𝜋𝑡subscript𝑎𝑡\displaystyle=n_{I}-1-\sum_{j=1}^{t-1}\mathds{1}_{\pi_{j}=\pi_{t}}=a_{t}. (A.38)

Therefore, from (A.11), (A) and (A), we have

ωtsubscriptsuperscript𝜔𝑡\displaystyle\omega^{\prime}_{t} =atβI+(ntat)βcabsentsubscriptsuperscript𝑎𝑡subscript𝛽𝐼𝑛𝑡subscriptsuperscript𝑎𝑡subscript𝛽𝑐\displaystyle=a^{\prime}_{t}\beta_{I}+(n-t-a^{\prime}_{t})\beta_{c}
=at+1βi+(ntat+1)βc=ωt+1+βc.absentsubscript𝑎𝑡1subscript𝛽𝑖𝑛𝑡subscript𝑎𝑡1subscript𝛽𝑐subscript𝜔𝑡1subscript𝛽𝑐\displaystyle=a_{t+1}\beta_{i}+(n-t-a_{t+1})\beta_{c}=\omega_{t+1}+\beta_{c}.

Similarly, ωt+1=ωtβcsubscriptsuperscript𝜔𝑡1subscript𝜔𝑡subscript𝛽𝑐\omega^{\prime}_{t+1}=\omega_{t}-\beta_{c} holds. This proves (A.36). Thus, from (A.33) and (A.36),

L(𝒔,𝝅)=i=1kmin{α,ωi}=i{t,t+1}min{α,ωi}+ωt+α,𝐿𝒔𝝅superscriptsubscript𝑖1𝑘𝛼subscript𝜔𝑖subscript𝑖𝑡𝑡1𝛼subscript𝜔𝑖subscript𝜔𝑡𝛼L(\bm{s},\bm{\pi})=\sum_{i=1}^{k}\min\{\alpha,\omega_{i}\}=\sum_{i\notin\{t,t+1\}}\min\{\alpha,\omega_{i}\}+\omega_{t}+\alpha,
L(𝒔,𝝅)𝐿𝒔superscript𝝅\displaystyle L(\bm{s},\bm{\pi}^{\prime}) =i=1kmin{α,ωi}absentsuperscriptsubscript𝑖1𝑘𝛼superscriptsubscript𝜔𝑖\displaystyle=\sum_{i=1}^{k}\min\{\alpha,\omega_{i}^{\prime}\}
=i{t,t+1}min{α,ωi}+α+(ωtβc)absentsubscript𝑖𝑡𝑡1𝛼superscriptsubscript𝜔𝑖𝛼subscript𝜔𝑡subscript𝛽𝑐\displaystyle=\sum_{i\notin\{t,t+1\}}\min\{\alpha,\omega_{i}^{\prime}\}+\alpha+(\omega_{t}-\beta_{c})
=i{t,t+1}min{α,ωi}+α+(ωtβc).absentsubscript𝑖𝑡𝑡1𝛼subscript𝜔𝑖𝛼subscript𝜔𝑡subscript𝛽𝑐\displaystyle=\sum_{i\notin\{t,t+1\}}\min\{\alpha,\omega_{i}\}+\alpha+(\omega_{t}-\beta_{c}).

Therefore, L(𝒔,𝝅)>L(𝒔,𝝅)𝐿𝒔𝝅𝐿𝒔superscript𝝅L(\bm{s},\bm{\pi})>L(\bm{s},\bm{\pi}^{\prime}) holds for βc>0subscript𝛽𝑐0\beta_{c}>0. In other words, if πΠpc𝜋superscriptsubscriptΠ𝑝𝑐\pi\in\Pi_{p}^{c}, then πΠmc𝜋superscriptsubscriptΠ𝑚𝑐\pi\in\Pi_{m}^{c}. This proves that ΠpcΠmcsuperscriptsubscriptΠ𝑝𝑐superscriptsubscriptΠ𝑚𝑐\Pi_{p}^{c}\subseteq\Pi_{m}^{c} holds for βc0subscript𝛽𝑐0\beta_{c}\neq 0.

Similarly, ΠpcΠrcsuperscriptsubscriptΠ𝑝𝑐superscriptsubscriptΠ𝑟𝑐\Pi_{p}^{c}\subseteq\Pi_{r}^{c} can be proved as follows. For the pre-defined ordering vectors 𝝅𝝅\bm{\pi} and 𝝅superscript𝝅bold-′\bm{\pi^{\prime}}, we have St(𝝅)=i=1t1ωi+ωtsubscript𝑆𝑡𝝅superscriptsubscript𝑖1𝑡1subscript𝜔𝑖subscript𝜔𝑡S_{t}(\bm{\pi})=\sum_{i=1}^{t-1}\omega_{i}+\omega_{t} and St(𝝅)=i=1t1ωi+ωt+1+βcsubscript𝑆𝑡superscript𝝅bold-′superscriptsubscript𝑖1𝑡1subscript𝜔𝑖subscript𝜔𝑡1subscript𝛽𝑐S_{t}(\bm{\pi^{\prime}})=\sum_{i=1}^{t-1}\omega_{i}+\omega_{t+1}+\beta_{c}. Using (A.33), we have St(𝝅)<St(𝝅)subscript𝑆𝑡𝝅subscript𝑆𝑡superscript𝝅bold-′S_{t}(\bm{\pi})<S_{t}(\bm{\pi^{\prime}}), so that 𝝅𝝅\bm{\pi} cannot be a running-sum maximizer. Therefore, ΠpcΠrcsuperscriptsubscriptΠ𝑝𝑐superscriptsubscriptΠ𝑟𝑐\Pi_{p}^{c}\subseteq\Pi_{r}^{c} holds.

Step 1-2. Prove that L(𝒔,𝝅)L(𝒔,𝝅)𝐿𝒔superscript𝝅𝐿𝒔𝝅L(\bm{s,\pi^{*}})\leq L(\bm{s,\pi}), 𝝅Πr,𝝅ΠpΠrcformulae-sequencefor-allsuperscript𝝅subscriptΠ𝑟for-all𝝅subscriptΠ𝑝superscriptsubscriptΠ𝑟𝑐\forall\bm{\pi}^{*}\in\Pi_{r},\forall\bm{\pi}\in\Pi_{p}\cap\Pi_{r}^{c} :

Consider arbitrary 𝝅Πrsuperscript𝝅subscriptΠ𝑟\bm{\pi}^{*}\in\Pi_{r} and 𝝅ΠpΠrc𝝅subscriptΠ𝑝superscriptsubscriptΠ𝑟𝑐\bm{\pi}\in\Pi_{p}\cap\Pi_{r}^{c}. For i[k]𝑖delimited-[]𝑘i\in[k], let ωisuperscriptsubscript𝜔𝑖\omega_{i}^{*} and ωisubscript𝜔𝑖\omega_{i} be short-hand notations for ωi(𝝅)subscript𝜔𝑖superscript𝝅\omega_{i}(\bm{\pi}^{*}) and ωi(𝝅)subscript𝜔𝑖𝝅\omega_{i}(\bm{\pi}), respectively. Note that from Proposition 3,

i=1kωi=i=1kωi.superscriptsubscript𝑖1𝑘subscript𝜔𝑖superscriptsubscript𝑖1𝑘superscriptsubscript𝜔𝑖\sum_{i=1}^{k}\omega_{i}=\sum_{i=1}^{k}\omega_{i}^{*}. (A.39)

Let

t𝑡\displaystyle t =max{i[k]:wiα},absent:𝑖delimited-[]𝑘subscript𝑤𝑖𝛼\displaystyle=\max\{i\in[k]:w_{i}\geq\alpha\},
tsuperscript𝑡\displaystyle t^{*} =max{i[k]:wiα}absent:𝑖delimited-[]𝑘superscriptsubscript𝑤𝑖𝛼\displaystyle=\max\{i\in[k]:w_{i}^{*}\geq\alpha\} (A.40)

Then, from (A.29),

i=1tωii=1tωi.superscriptsubscript𝑖1𝑡subscript𝜔𝑖superscriptsubscript𝑖1𝑡superscriptsubscript𝜔𝑖\sum_{i=1}^{t}\omega_{i}\leq\sum_{i=1}^{t}\omega_{i}^{*}. (A.41)

Combining with (A.39), we obtain

i=t+1k(ωiωi)0.superscriptsubscript𝑖𝑡1𝑘subscript𝜔𝑖superscriptsubscript𝜔𝑖0\sum_{i=t+1}^{k}(\omega_{i}-\omega_{i}^{*})\geq 0. (A.42)

Note that from the result of Step 1-1, both 𝝅𝝅\bm{\pi} and 𝝅superscript𝝅\bm{\pi}^{*} are in ΠpsubscriptΠ𝑝\Pi_{p}. Therefore, ωiαsubscript𝜔𝑖𝛼\omega_{i}\geq\alpha for i[t]𝑖delimited-[]𝑡i\in[t]. Similarly, ωiαsuperscriptsubscript𝜔𝑖𝛼\omega_{i}^{*}\geq\alpha for i[t]𝑖delimited-[]superscript𝑡i\in[t^{*}]. Therefore, (A.20) can be expressed as

L(𝒔,𝝅)𝐿𝒔𝝅\displaystyle L(\bm{s},\bm{\pi}) =i=1kmin{α,ωi}=i=1tα+i=t+1kωi,absentsuperscriptsubscript𝑖1𝑘𝛼subscript𝜔𝑖superscriptsubscript𝑖1𝑡𝛼superscriptsubscript𝑖𝑡1𝑘subscript𝜔𝑖\displaystyle=\sum_{i=1}^{k}\min\{\alpha,\omega_{i}\}=\sum_{i=1}^{t}\alpha+\sum_{i=t+1}^{k}\omega_{i}, (A.43)
L(𝒔,𝝅)𝐿𝒔superscript𝝅\displaystyle L(\bm{s},\bm{\pi}^{*}) =i=1kmin{α,ωi}=i=1tα+i=t+1kωi.absentsuperscriptsubscript𝑖1𝑘𝛼superscriptsubscript𝜔𝑖superscriptsubscript𝑖1superscript𝑡𝛼superscriptsubscript𝑖superscript𝑡1𝑘superscriptsubscript𝜔𝑖\displaystyle=\sum_{i=1}^{k}\min\{\alpha,\omega_{i}^{*}\}=\sum_{i=1}^{t^{*}}\alpha+\sum_{i=t^{*}+1}^{k}\omega_{i}^{*}. (A.44)

If t=t𝑡superscript𝑡t=t^{*}, then we have

L(𝒔,𝝅)L(𝒔,𝝅)=i=t+1k(ωiωi)0𝐿𝒔𝝅𝐿𝒔superscript𝝅superscriptsubscript𝑖𝑡1𝑘subscript𝜔𝑖superscriptsubscript𝜔𝑖0L(\bm{s,\pi})-L(\bm{s,\pi^{*}})=\sum_{i=t+1}^{k}(\omega_{i}-\omega_{i}^{*})\geq 0

from (A.42). If t>t𝑡superscript𝑡t>t^{*}, we get

L(𝒔,𝝅)L(𝒔,𝝅)=i=t+1t(αωi)+i=t+1k(ωiωi).𝐿𝒔𝝅𝐿𝒔superscript𝝅superscriptsubscript𝑖superscript𝑡1𝑡𝛼superscriptsubscript𝜔𝑖superscriptsubscript𝑖𝑡1𝑘subscript𝜔𝑖superscriptsubscript𝜔𝑖L(\bm{s,\pi})-L(\bm{s,\pi^{*}})=\sum_{i=t^{*}+1}^{t}(\alpha-\omega_{i}^{*})+\sum_{i=t+1}^{k}(\omega_{i}-\omega_{i}^{*}).

From (A), ωi<αsuperscriptsubscript𝜔𝑖𝛼\omega_{i}^{*}<\alpha holds for i>t𝑖superscript𝑡i>t^{*}. Therefore,

L(𝒔,𝝅)L(𝒔,𝝅)>i=t+1k(ωiωi)0𝐿𝒔𝝅𝐿𝒔superscript𝝅superscriptsubscript𝑖𝑡1𝑘subscript𝜔𝑖superscriptsubscript𝜔𝑖0L(\bm{s,\pi})-L(\bm{s,\pi^{*}})>\sum_{i=t+1}^{k}(\omega_{i}-\omega_{i}^{*})\geq 0

from (A.42). In the case of t<t𝑡superscript𝑡t<t^{*}, define Δ=i=1t(ωiα)Δsuperscriptsubscript𝑖1𝑡subscript𝜔𝑖𝛼\Delta=\sum_{i=1}^{t}(\omega_{i}-\alpha) and Δ=i=1t(ωiα)superscriptΔsuperscriptsubscript𝑖1𝑡superscriptsubscript𝜔𝑖𝛼\Delta^{*}=\sum_{i=1}^{t}(\omega_{i}^{*}-\alpha). From (A.43),

L(𝒔,𝝅)=i=1tα+i=t+1kωi=(i=1kωi)Δ.𝐿𝒔𝝅superscriptsubscript𝑖1𝑡𝛼superscriptsubscript𝑖𝑡1𝑘subscript𝜔𝑖superscriptsubscript𝑖1𝑘subscript𝜔𝑖ΔL(\bm{s,\pi})=\sum_{i=1}^{t}\alpha+\sum_{i=t+1}^{k}\omega_{i}=(\sum_{i=1}^{k}\omega_{i})-\Delta.

Similarly, from (A.44),

L(𝒔,𝝅)𝐿𝒔superscript𝝅\displaystyle L(\bm{s,\pi^{*}}) =i=1kωii=1t(ωiα)absentsuperscriptsubscript𝑖1𝑘superscriptsubscript𝜔𝑖superscriptsubscript𝑖1superscript𝑡superscriptsubscript𝜔𝑖𝛼\displaystyle=\sum_{i=1}^{k}\omega_{i}^{*}-\sum_{i=1}^{t^{*}}(\omega_{i}^{*}-\alpha)
=i=1kωiΔi=t+1t(ωiα)i=1kωiΔabsentsuperscriptsubscript𝑖1𝑘superscriptsubscript𝜔𝑖superscriptΔsuperscriptsubscript𝑖𝑡1superscript𝑡superscriptsubscript𝜔𝑖𝛼superscriptsubscript𝑖1𝑘superscriptsubscript𝜔𝑖superscriptΔ\displaystyle=\sum_{i=1}^{k}\omega_{i}^{*}-\Delta^{*}-\sum_{i=t+1}^{t^{*}}(\omega_{i}^{*}-\alpha)\leq\sum_{i=1}^{k}\omega_{i}^{*}-\Delta^{*}

where the last inequality is from (A). Combined with (A.39) and (A.41), we obtain

L(𝒔,𝝅)L(𝒔,𝝅)ΔΔ0.𝐿𝒔𝝅𝐿𝒔superscript𝝅superscriptΔΔ0L(\bm{s,\pi})-L(\bm{s,\pi^{*}})\geq\Delta^{*}-\Delta\geq 0.

In summary, L(𝒔,𝝅)L(𝒔,𝝅)𝐿𝒔superscript𝝅𝐿𝒔𝝅L(\bm{s,\pi^{*}})\leq L(\bm{s,\pi}) irrespective of the t,t𝑡superscript𝑡t,t^{*} values, which completes the proof for Step 1-2. From the results of Step 1-1 and Step 1-2, the relationship between the sets can be depicted as in Fig. 24.

Refer to caption
Figure 24: Relationship between sets

Consider 𝝅0Πrsubscript𝝅0subscriptΠ𝑟\bm{\pi}_{0}\in\Pi_{r} and 𝝅ΠmΠrcsuperscript𝝅subscriptΠ𝑚superscriptsubscriptΠ𝑟𝑐\bm{\pi}^{*}\in\Pi_{m}\cap\Pi_{r}^{c}. Then, L(𝒔,𝝅0)L(𝒔,𝝅)𝐿𝒔subscript𝝅0𝐿𝒔superscript𝝅L(\bm{s},\bm{\pi}_{0})\leq L(\bm{s},\bm{\pi}^{*}) from the result of Step 1-2. Based on the definition of ΠmsubscriptΠ𝑚\Pi_{m} in (A.30), we can write L(𝒔,𝝅𝟎)L(𝒔,𝝅)𝐿𝒔subscript𝝅0𝐿𝒔𝝅L(\bm{s},\bm{\pi_{0}})\leq L(\bm{s},\bm{\pi}) for every 𝝅Π(𝒔).𝝅Π𝒔\bm{\pi}\in\Pi(\bm{s}). In other words, π0Πmsubscript𝜋0subscriptΠ𝑚\pi_{0}\in\Pi_{m} holds for arbitrary π0Πrsubscript𝜋0subscriptΠ𝑟\pi_{0}\in\Pi_{r}. Therefore, ΠrΠmsubscriptΠ𝑟subscriptΠ𝑚\Pi_{r}\subseteq\Pi_{m} holds.

Step 2. Prove 𝝅vΠrsubscript𝝅𝑣subscriptΠ𝑟\bm{\pi}_{v}\in\Pi_{r}:

Refer to caption
Figure 25: Set of s1subscript𝑠1s_{1} lines where (i,ωi)𝑖subscript𝜔𝑖(i,\omega_{i}) points can position

For a given selection vector 𝒔=[s1,,sL]𝒔subscript𝑠1subscript𝑠𝐿\bm{s}=[s_{1},\cdots,s_{L}], consider an arbitrary ordering vector 𝝅=[π1,,πk]Π(𝒔)𝝅subscript𝜋1subscript𝜋𝑘Π𝒔\bm{\pi}=[\pi_{1},\cdots,\pi_{k}]\in\Pi(\bm{s}). The corresponding ωi(𝝅)subscript𝜔𝑖𝝅\omega_{i}(\bm{\pi}) defined in (A.21) is written as

wi=(nIj)βI+(ninI+j)βcsubscript𝑤𝑖subscript𝑛𝐼𝑗subscript𝛽𝐼𝑛𝑖subscript𝑛𝐼𝑗subscript𝛽𝑐w_{i}=(n_{I}-j)\beta_{I}+(n-i-n_{I}+j)\beta_{c} (A.45)

where j=t=1i𝟙πt=πi𝑗superscriptsubscript𝑡1𝑖subscript1subscript𝜋𝑡subscript𝜋𝑖j=\sum_{t=1}^{i}\mathds{1}_{\pi_{t}=\pi_{i}}.

Consider a set of lines {lj}j=1nIsuperscriptsubscriptsubscript𝑙𝑗𝑗1subscript𝑛𝐼\{l_{j}\}_{j=1}^{n_{I}}, where line ljsubscript𝑙𝑗l_{j} represents an equation: wi=(nIj)βI+(ninI+j)βcsubscript𝑤𝑖subscript𝑛𝐼𝑗subscript𝛽𝐼𝑛𝑖subscript𝑛𝐼𝑗subscript𝛽𝑐w_{i}=(n_{I}-j)\beta_{I}+(n-i-n_{I}+j)\beta_{c}. Since we assume βIβcsubscript𝛽𝐼subscript𝛽𝑐\beta_{I}\geq\beta_{c}, these lines can be illustrated as in Fig. 25. For a given 𝝅𝝅\bm{\pi}, consider marking a (i,ωi)𝑖subscript𝜔𝑖(i,\omega_{i}) point for i[k]𝑖delimited-[]𝑘i\in[k]. Note that the (i,ωi)𝑖subscript𝜔𝑖(i,\omega_{i}) point is on line ljsubscript𝑙𝑗l_{j} if and only if

j=t=1i𝟙πt=πi𝑗superscriptsubscript𝑡1𝑖subscript1subscript𝜋𝑡subscript𝜋𝑖j=\sum_{t=1}^{i}\mathds{1}_{\pi_{t}=\pi_{i}} (A.46)

where the summation term in (A.46) represents the number of occurrence of πisubscript𝜋𝑖\pi_{i} value in {πt}t=1isuperscriptsubscriptsubscript𝜋𝑡𝑡1𝑖\{\pi_{t}\}_{t=1}^{i}. For the example in Fig. 23, when 𝒔=[4,3,1]𝒔431\bm{s}=[4,3,1] and 𝝅=[1,2,3,1,2,1,2,1]Π(𝒔)𝝅12312121Π𝒔\bm{\pi}=[1,2,3,1,2,1,2,1]\in\Pi(\bm{s}), line l3subscript𝑙3l_{3} contains the point (i,ωi)=(6,ω6)𝑖subscript𝜔𝑖6subscript𝜔6(i,\omega_{i})=(6,\omega_{6}) since 3=t=16𝟙πt=π63superscriptsubscript𝑡16subscript1subscript𝜋𝑡subscript𝜋63=\sum_{t=1}^{6}\mathds{1}_{\pi_{t}=\pi_{6}}.

Recall that Il(𝝅)={i[k]:πi=l}subscript𝐼𝑙𝝅conditional-set𝑖delimited-[]𝑘subscript𝜋𝑖𝑙I_{l}(\bm{\pi})=\{i\in[k]:\pi_{i}=l\}, as defined in (A.26), where |Il(𝝅)|=slsubscript𝐼𝑙𝝅subscript𝑠𝑙|I_{l}(\bm{\pi})|=s_{l} holds l[L]for-all𝑙delimited-[]𝐿\forall l\in[L]. For j[nI]𝑗delimited-[]subscript𝑛𝐼j\in[n_{I}], consider l[L]𝑙delimited-[]𝐿l\in[L] with sljsubscript𝑠𝑙𝑗s_{l}\geq j. Let i0subscript𝑖0i_{0} be the jthsuperscript𝑗𝑡j^{th} smallest element in Il(𝝅)subscript𝐼𝑙𝝅I_{l}(\bm{\pi}). Then, j=t=1i0𝟙πt=l𝑗superscriptsubscript𝑡1subscript𝑖0subscript1subscript𝜋𝑡𝑙j=\sum_{t=1}^{i_{0}}\mathds{1}_{\pi_{t}=l} and πi0=lsubscript𝜋subscript𝑖0𝑙\pi_{i_{0}}=l hold. Thus, the (i0,ωi0)subscript𝑖0subscript𝜔subscript𝑖0(i_{0},\omega_{i_{0}}) point is on line ljsubscript𝑙𝑗l_{j}. Similarly, we can find

pj=l=1L𝟙sljsubscript𝑝𝑗superscriptsubscript𝑙1𝐿subscript1subscript𝑠𝑙𝑗p_{j}=\sum_{l=1}^{L}\mathds{1}_{s_{l}\geq j} (A.47)

points on line ljsubscript𝑙𝑗l_{j}, irrespective of the ordering vector 𝝅Π(𝒔)𝝅Π𝒔\bm{\pi}\in\Pi(\bm{s}). Note that

j=1nIpj=l=1Lj=1nI𝟙slj=l=1Lsl=k,superscriptsubscript𝑗1subscript𝑛𝐼subscript𝑝𝑗superscriptsubscript𝑙1𝐿superscriptsubscript𝑗1subscript𝑛𝐼subscript1subscript𝑠𝑙𝑗superscriptsubscript𝑙1𝐿subscript𝑠𝑙𝑘\sum_{j=1}^{n_{I}}p_{j}=\sum_{l=1}^{L}\sum_{j=1}^{n_{I}}\mathds{1}_{s_{l}\geq j}=\sum_{l=1}^{L}s_{l}=k, (A.48)

which confirms that Fig. 25 contains k𝑘k points. Moreover,

j[nI1],pjpj+1formulae-sequencefor-all𝑗delimited-[]subscript𝑛𝐼1subscript𝑝𝑗subscript𝑝𝑗1\forall j\in[n_{I}-1],\ p_{j}\geq p_{j+1} (A.49)

holds from the definition in (A.47).

Refer to caption
Figure 26: Optimal packing of k𝑘k points

In order to maximize the running sum St(𝝅)=i=1twi(𝝅)subscript𝑆𝑡𝝅superscriptsubscript𝑖1𝑡subscript𝑤𝑖𝝅S_{t}(\bm{\pi})=\sum_{i=1}^{t}w_{i}(\bm{\pi}) for every t𝑡t, the optimal ordering vector packs p1subscript𝑝1p_{1} points in the leftmost area (i=1,,p1𝑖1subscript𝑝1i=1,\cdots,p_{1}), pack p2subscript𝑝2p_{2} points in the leftmost remaining area (i=p1+1,,p1+p2𝑖subscript𝑝11subscript𝑝1subscript𝑝2i=p_{1}+1,\cdots,p_{1}+p_{2}), and so on. This packing method corresponds to Fig. 26.

Note that from the definition of pjsubscript𝑝𝑗p_{j} in (A.47) and Fig. 23, vertical ordering 𝝅vsubscript𝝅𝑣\bm{\pi}_{v} in Definition 3 first chooses p1subscript𝑝1p_{1} points on line l1subscript𝑙1l_{1}, then chooses p2subscript𝑝2p_{2} points on line l2subscript𝑙2l_{2}, and so on. Thus, 𝝅vsubscript𝝅𝑣\bm{\pi}_{v} achieves optimal packing in Fig. 26, which maximizes the running sum St(𝝅)subscript𝑆𝑡𝝅S_{t}(\bm{\pi}). Therefore, vertical ordering is a running sum maximizer, i.e., 𝝅vΠrsubscript𝝅𝑣subscriptΠ𝑟\bm{\pi}_{v}\in\Pi_{r}. Combining Steps 1 and 2, we conclude that 𝝅vsubscript𝝅𝑣\bm{\pi}_{v} minimizes L(𝒔,𝝅)𝐿𝒔𝝅L(\bm{s},\bm{\pi}) among 𝝅Π(𝒔)𝝅Π𝒔\bm{\pi}\in\Pi(\bm{s}) for arbitrary 𝒔S𝒔𝑆\bm{s}\in S. ∎

Now, we define a special selection vector called the horizontal selection vector, which is shown to be the optimal selection vector which minimizes L(𝒔,𝝅v)𝐿𝒔subscript𝝅𝑣L(\bm{s},\bm{\pi}_{v}).

Definition 4.

The horizontal selection vector 𝐬h=[s1,,sL]𝒮subscript𝐬subscript𝑠1subscript𝑠𝐿𝒮\bm{s}_{h}=[s_{1},\cdots,s_{L}]\in\mathcal{S} is defined as:

si={nI,iknI(kmodnI),i=knI+10i>knI+1.subscript𝑠𝑖casessubscript𝑛𝐼𝑖𝑘subscript𝑛𝐼𝑘modsubscript𝑛𝐼𝑖𝑘subscript𝑛𝐼10𝑖𝑘subscript𝑛𝐼1s_{i}=\begin{cases}n_{I},&i\leq\lfloor\frac{k}{n_{I}}\rfloor\\ (k\ \mathrm{mod}\ n_{I}),&i=\lfloor\frac{k}{n_{I}}\rfloor+1\\ 0&i>\lfloor\frac{k}{n_{I}}\rfloor+1.\end{cases}
Refer to caption
Figure 27: The optimal selection vector 𝒔hsubscript𝒔\bm{s}_{h} and the optimal ordering vector 𝝅vsubscript𝝅𝑣\bm{\pi}_{v} (for n=15,k=8,L=3formulae-sequence𝑛15formulae-sequence𝑘8𝐿3n=15,k=8,L=3 case)

The graphical illustration of the horizontal selection vector is on the left side of Fig. 27, in the case of n=15,k=8,L=3formulae-sequence𝑛15formulae-sequence𝑘8𝐿3n=15,k=8,L=3. The following Lemma states that the horizontal selection vector minimizes L(𝒔,𝝅v)𝐿𝒔subscript𝝅𝑣L(\bm{s},\bm{\pi}_{v}).

Lemma 4.

Consider applying the vertical ordering vector 𝛑vsubscript𝛑𝑣\bm{\pi}_{v}. Then, the horizontal selection vector 𝐬hsubscript𝐬\bm{s}_{h} minimizes the lower bound L(𝐬,𝛑)𝐿𝐬𝛑L(\bm{s},\bm{\pi}) on the min-cut. In other words, L(𝐬h,𝛑v)L(𝐬,𝛑v)𝐬𝒮𝐿subscript𝐬subscript𝛑𝑣𝐿𝐬subscript𝛑𝑣for-all𝐬𝒮L(\bm{s}_{h},\bm{\pi}_{v})\leq L(\bm{s},\bm{\pi}_{v})\ \forall\bm{s}\in\mathcal{S}.

Proof.

From the proof of Lemma 3, the optimal ordering vector turns out to be the vertical ordering vector, where the corresponding ωi(𝝅v)subscript𝜔𝑖subscript𝝅𝑣\omega_{i}(\bm{\pi}_{v}) sequence is illustrated in Fig. 26. Depending on the selection vector 𝒔=[s1,,sL]𝒔subscript𝑠1subscript𝑠𝐿\bm{s}=[s_{1},\cdots,s_{L}], the number pjsubscript𝑝𝑗p_{j} of points on each line ljsubscript𝑙𝑗l_{j} changes.

Consider an arbitrary selection vector 𝒔𝒔\bm{s}. Define a point vector 𝒑(𝒔)=[p1,,pnI]𝒑𝒔subscript𝑝1subscript𝑝subscript𝑛𝐼\bm{p(s)}=[p_{1},\cdots,p_{n_{I}}] where pjsubscript𝑝𝑗p_{j} is the number of points on ljsubscript𝑙𝑗l_{j}, as defined in (A.47). Similarly, define 𝒑(𝒔𝒉)=[p1,,pnI]𝒑subscript𝒔𝒉superscriptsubscript𝑝1superscriptsubscript𝑝subscript𝑛𝐼\bm{p(s_{h})}=[p_{1}^{*},\cdots,p_{n_{I}}^{*}]. Using Definition 4 and (A.47), we have

pj={knI+1,j(kmodnI)knIotherwisesuperscriptsubscript𝑝𝑗cases𝑘subscript𝑛𝐼1𝑗𝑘modsubscript𝑛𝐼𝑘subscript𝑛𝐼𝑜𝑡𝑒𝑟𝑤𝑖𝑠𝑒p_{j}^{*}=\begin{cases}\lfloor\frac{k}{n_{I}}\rfloor+1,&\ j\leq(k\ \mathrm{mod}\ n_{I})\\ \lfloor\frac{k}{n_{I}}\rfloor&\ otherwise\end{cases} (A.50)

Now, we prove

t[nI],j=1tpjj=1tpj.formulae-sequencefor-all𝑡delimited-[]subscript𝑛𝐼superscriptsubscript𝑗1𝑡superscriptsubscript𝑝𝑗superscriptsubscript𝑗1𝑡subscript𝑝𝑗\forall t\in[n_{I}],\ \sum_{j=1}^{t}p_{j}^{*}\leq\sum_{j=1}^{t}p_{j}. (A.51)

The proof is divided into two steps: base case and inductive step.

Base Case: We wish to prove that p1p1superscriptsubscript𝑝1subscript𝑝1p_{1}^{*}\leq p_{1}. Suppose p1>p1superscriptsubscript𝑝1subscript𝑝1p_{1}^{*}>p_{1} (i.e., p1p11subscript𝑝1superscriptsubscript𝑝11p_{1}\leq p_{1}^{*}-1). Then,

l=1nIpll=1nIp1l=1nI(p11)superscriptsubscript𝑙1subscript𝑛𝐼subscript𝑝𝑙superscriptsubscript𝑙1subscript𝑛𝐼subscript𝑝1superscriptsubscript𝑙1subscript𝑛𝐼superscriptsubscript𝑝11\sum_{l=1}^{n_{I}}p_{l}\leq\sum_{l=1}^{n_{I}}p_{1}\leq\sum_{l=1}^{n_{I}}(p_{1}^{*}-1) (A.52)

where the first inequality is from (A.49). Note that if (kmodnI)=0𝑘modsubscript𝑛𝐼0(k\ \mathrm{mod}\ n_{I})=0, then

l=1nIpl=l=1nIp1>l=1nI(p11).superscriptsubscript𝑙1subscript𝑛𝐼superscriptsubscript𝑝𝑙superscriptsubscript𝑙1subscript𝑛𝐼superscriptsubscript𝑝1superscriptsubscript𝑙1subscript𝑛𝐼superscriptsubscript𝑝11\sum_{l=1}^{n_{I}}p_{l}^{*}=\sum_{l=1}^{n_{I}}p_{1}^{*}>\sum_{l=1}^{n_{I}}(p_{1}^{*}-1). (A.53)

Otherwise,

l=1nIplsuperscriptsubscript𝑙1subscript𝑛𝐼superscriptsubscript𝑝𝑙\displaystyle\sum_{l=1}^{n_{I}}p_{l}^{*} =l=1(kmodnI)pl+l=(kmodnI)+1nIplabsentsuperscriptsubscript𝑙1𝑘modsubscript𝑛𝐼superscriptsubscript𝑝𝑙superscriptsubscript𝑙𝑘modsubscript𝑛𝐼1subscript𝑛𝐼superscriptsubscript𝑝𝑙\displaystyle=\sum_{l=1}^{(k\ \mathrm{mod}\ n_{I})}p_{l}^{*}+\sum_{l=(k\ \mathrm{mod}\ n_{I})+1}^{n_{I}}p_{l}^{*}
=l=1(kmodnI)(knI+1)+l=(kmodnI)+1nIknIabsentsuperscriptsubscript𝑙1𝑘modsubscript𝑛𝐼𝑘subscript𝑛𝐼1superscriptsubscript𝑙𝑘modsubscript𝑛𝐼1subscript𝑛𝐼𝑘subscript𝑛𝐼\displaystyle=\sum_{l=1}^{(k\ \mathrm{mod}\ n_{I})}(\lfloor\frac{k}{n_{I}}\rfloor+1)+\sum_{l=(k\ \mathrm{mod}\ n_{I})+1}^{n_{I}}\lfloor\frac{k}{n_{I}}\rfloor
>l=1nIknI=l=1nI(p11).absentsuperscriptsubscript𝑙1subscript𝑛𝐼𝑘subscript𝑛𝐼superscriptsubscript𝑙1subscript𝑛𝐼superscriptsubscript𝑝11\displaystyle>\sum_{l=1}^{n_{I}}\lfloor\frac{k}{n_{I}}\rfloor=\sum_{l=1}^{n_{I}}(p_{1}^{*}-1). (A.54)

Therefore, combining (A.52), (A.53) and (A.54) results in l=1nIpl<l=1nIpl=ksuperscriptsubscript𝑙1subscript𝑛𝐼subscript𝑝𝑙superscriptsubscript𝑙1subscript𝑛𝐼superscriptsubscript𝑝𝑙𝑘\sum_{l=1}^{n_{I}}p_{l}<\sum_{l=1}^{n_{I}}p_{l}^{*}=k, which contradicts (A.48). Therefore, p1p1superscriptsubscript𝑝1subscript𝑝1p_{1}^{*}\leq p_{1} holds.

Inductive Step: Assume that l=1l0pll=1l0plsuperscriptsubscript𝑙1subscript𝑙0superscriptsubscript𝑝𝑙superscriptsubscript𝑙1subscript𝑙0subscript𝑝𝑙\sum_{l=1}^{l_{0}}p_{l}^{*}\leq\sum_{l=1}^{l_{0}}p_{l} for arbitrary lo[nI1]subscript𝑙𝑜delimited-[]subscript𝑛𝐼1l_{o}\in[n_{I}-1]. Now we prove that l=1l0+1pll=1l0+1plsuperscriptsubscript𝑙1subscript𝑙01superscriptsubscript𝑝𝑙superscriptsubscript𝑙1subscript𝑙01subscript𝑝𝑙\sum_{l=1}^{l_{0}+1}p_{l}^{*}\leq\sum_{l=1}^{l_{0}+1}p_{l} holds. Suppose not. Then,

pl0+1Θ1pl0+1superscriptsubscript𝑝subscript𝑙01Θ1subscript𝑝subscript𝑙01p_{l_{0}+1}^{*}-\Theta-1\geq p_{l_{0}+1} (A.55)

holds where

Θ=l=1l0(plpl).Θsuperscriptsubscript𝑙1subscript𝑙0subscript𝑝𝑙superscriptsubscript𝑝𝑙\Theta=\sum_{l=1}^{l_{0}}(p_{l}-p_{l}^{*}). (A.56)

Using (A.49) and (A.55), we have

l=l0+1nIplsuperscriptsubscript𝑙subscript𝑙01subscript𝑛𝐼subscript𝑝𝑙\displaystyle\sum_{l=l_{0}+1}^{n_{I}}p_{l} l=l0+1nIpl0+1l=l0+1nI(pl0+11Θ)absentsuperscriptsubscript𝑙subscript𝑙01subscript𝑛𝐼subscript𝑝subscript𝑙01superscriptsubscript𝑙subscript𝑙01subscript𝑛𝐼superscriptsubscript𝑝subscript𝑙011Θ\displaystyle\leq\sum_{l=l_{0}+1}^{n_{I}}p_{l_{0}+1}\leq\sum_{l=l_{0}+1}^{n_{I}}(p_{l_{0}+1}^{*}-1-\Theta)
l=l0+1nI(pl0+11)Θabsentsuperscriptsubscript𝑙subscript𝑙01subscript𝑛𝐼superscriptsubscript𝑝subscript𝑙011Θ\displaystyle\leq\sum_{l=l_{0}+1}^{n_{I}}(p_{l_{0}+1}^{*}-1)-\Theta (A.57)

where equality holds for the last inequality iff l0=nI1subscript𝑙0subscript𝑛𝐼1l_{0}=n_{I}-1. Using analysis similar to (A.53) and (A.54) for the base case, we can find that l=l0+1nI(pl0+11)<l=l0+1nIpl.superscriptsubscript𝑙subscript𝑙01subscript𝑛𝐼superscriptsubscript𝑝subscript𝑙011superscriptsubscript𝑙subscript𝑙01subscript𝑛𝐼superscriptsubscript𝑝𝑙\sum_{l=l_{0}+1}^{n_{I}}(p_{l_{0}+1}^{*}-1)<\sum_{l=l_{0}+1}^{n_{I}}p_{l}^{*}. Combining with (A.57), we get

l=l0+1nIpl<l=l0+1nIplΘ.superscriptsubscript𝑙subscript𝑙01subscript𝑛𝐼subscript𝑝𝑙superscriptsubscript𝑙subscript𝑙01subscript𝑛𝐼superscriptsubscript𝑝𝑙Θ\sum_{l=l_{0}+1}^{n_{I}}p_{l}<\sum_{l=l_{0}+1}^{n_{I}}p_{l}^{*}-\Theta. (A.58)

Equations (A.56) and (A.58) imply l=1nIpl<l=1nIpl=k,superscriptsubscript𝑙1subscript𝑛𝐼subscript𝑝𝑙superscriptsubscript𝑙1subscript𝑛𝐼superscriptsubscript𝑝𝑙𝑘\sum_{l=1}^{n_{I}}p_{l}<\sum_{l=1}^{n_{I}}p_{l}^{*}=k, which contradicts (A.48). Therefore, (A.51) holds.

Now define

fisubscript𝑓𝑖\displaystyle f_{i} =min{s[nI]:l=1spli}absent:𝑠delimited-[]subscript𝑛𝐼superscriptsubscript𝑙1𝑠subscript𝑝𝑙𝑖\displaystyle=\min\{s\in[n_{I}]:\sum_{l=1}^{s}p_{l}\geq i\} (A.59)
hisubscript𝑖\displaystyle h_{i} =min{s[nI]:l=1spli}absent:𝑠delimited-[]subscript𝑛𝐼superscriptsubscript𝑙1𝑠superscriptsubscript𝑝𝑙𝑖\displaystyle=\min\{s\in[n_{I}]:\sum_{l=1}^{s}p_{l}^{*}\geq i\} (A.60)

for i[k]𝑖delimited-[]𝑘i\in[k]. Then,

i[k],hififormulae-sequencefor-all𝑖delimited-[]𝑘subscript𝑖subscript𝑓𝑖\forall i\in[k],\ h_{i}\geq f_{i} (A.61)

holds directly from (A.51). Note that since pisuperscriptsubscript𝑝𝑖p_{i}^{*} in (A.50) is identical to gisubscript𝑔𝑖g_{i} in (7), hisubscript𝑖h_{i} can be written as

hi=min{s[nI]:l=1sgli}subscript𝑖:𝑠delimited-[]subscript𝑛𝐼superscriptsubscript𝑙1𝑠subscript𝑔𝑙𝑖h_{i}=\min\{s\in[n_{I}]:\sum_{l=1}^{s}g_{l}\geq i\} (A.62)

Consider 𝝅v(𝒔)subscript𝝅𝑣𝒔\bm{\pi}_{v}(\bm{s}), the vertical ordering vector for a given selection vector 𝒔𝒔\bm{s}. Recall that as in Fig. 26, vertical ordering packs the leftmost p1subscript𝑝1p_{1} points on line l1subscript𝑙1l_{1}, packs the next p2subscript𝑝2p_{2} points on line l2subscript𝑙2l_{2}, and so on. Using (A.45), we can write

ωi={(nI1)βI+(ninI+1)βc,if i[p1](nI2)βI+(ninI+2)βc,if ip1[p2]0βI+(ni)βc,if it=1nI1pt[pnI]subscript𝜔𝑖casessubscript𝑛𝐼1subscript𝛽𝐼𝑛𝑖subscript𝑛𝐼1subscript𝛽𝑐if 𝑖delimited-[]subscript𝑝1subscript𝑛𝐼2subscript𝛽𝐼𝑛𝑖subscript𝑛𝐼2subscript𝛽𝑐if 𝑖subscript𝑝1delimited-[]subscript𝑝2otherwise0subscript𝛽𝐼𝑛𝑖subscript𝛽𝑐if 𝑖superscriptsubscript𝑡1subscript𝑛𝐼1subscript𝑝𝑡otherwiseabsentdelimited-[]subscript𝑝subscript𝑛𝐼\displaystyle\omega_{i}=\begin{cases}(n_{I}-1)\beta_{I}+(n-i-n_{I}+1)\beta_{c},\quad&\text{if }i\in[p_{1}]\\ (n_{I}-2)\beta_{I}+(n-i-n_{I}+2)\beta_{c},\quad&\text{if }i-p_{1}\in[p_{2}]\\ \quad\quad\vdots\\ 0\cdot\beta_{I}+(n-i)\beta_{c},&\text{if }i-\sum_{t=1}^{n_{I}-1}p_{t}\\ &\quad\quad\in[p_{n_{I}}]\end{cases}

Therefore, using (A.59), we further write

ωi(𝝅v(𝒔))=(nIfi)βI+(ninI+fi)βc.subscript𝜔𝑖subscript𝝅𝑣𝒔subscript𝑛𝐼subscript𝑓𝑖subscript𝛽𝐼𝑛𝑖subscript𝑛𝐼subscript𝑓𝑖subscript𝛽𝑐\omega_{i}(\bm{\pi}_{v}(\bm{s}))=(n_{I}-f_{i})\beta_{I}+(n-i-n_{I}+f_{i})\beta_{c}. (A.63)

Similarly,

ωi(𝝅v(𝒔h))=(nIhi)βI+(ninI+hi)βc.subscript𝜔𝑖subscript𝝅𝑣subscript𝒔subscript𝑛𝐼subscript𝑖subscript𝛽𝐼𝑛𝑖subscript𝑛𝐼subscript𝑖subscript𝛽𝑐\omega_{i}(\bm{\pi}_{v}(\bm{s}_{h}))=(n_{I}-h_{i})\beta_{I}+(n-i-n_{I}+h_{i})\beta_{c}. (A.64)

Combining (A.61), (A.63) and (A.64), we have

ωi(𝝅v(𝒔h))ωi(𝝅v(𝒔))i=1,,k,𝒔𝒮,formulae-sequencesubscript𝜔𝑖subscript𝝅𝑣subscript𝒔subscript𝜔𝑖subscript𝝅𝑣𝒔for-all𝑖1𝑘for-all𝒔𝒮\omega_{i}(\bm{\pi}_{v}(\bm{s}_{h}))\leq\omega_{i}(\bm{\pi}_{v}(\bm{s}))\ \forall i=1,\cdots,k,\forall\bm{s}\in\mathcal{S},

since we assume βIβcsubscript𝛽𝐼subscript𝛽𝑐\beta_{I}\geq\beta_{c}. Therefore, combining with (A.20), we conclude that L(𝒔h,𝝅v)L(𝒔,𝝅v)𝐿subscript𝒔subscript𝝅𝑣𝐿𝒔subscript𝝅𝑣L(\bm{s}_{h},\bm{\pi}_{v})\leq L(\bm{s},\bm{\pi}_{v}) for arbitrary 𝒔𝒮𝒔𝒮\bm{s}\in\mathcal{S}, which completes the proof of Lemma 4. ∎

From Lemmas 3 and 4, we have

𝒔𝒮,𝝅Π(𝒔),L(𝒔h,𝝅v)L(𝒔,𝝅).formulae-sequencefor-all𝒔𝒮formulae-sequencefor-all𝝅Π𝒔𝐿subscript𝒔subscript𝝅𝑣𝐿𝒔𝝅\forall\bm{s}\in\mathcal{S},\forall\bm{\pi}\in\Pi(\bm{s}),\ L(\bm{s}_{h},\bm{\pi}_{v})\leq L(\bm{s},\bm{\pi}).

All that remains is to compute L(𝒔h,𝝅v)𝐿subscript𝒔subscript𝝅𝑣L(\bm{s}_{h},\bm{\pi}_{v}) and check that it is identical to (A.1).

From (A.20), L(𝒔h,𝝅v)𝐿subscript𝒔subscript𝝅𝑣L(\bm{s}_{h},\bm{\pi}_{v}) can be written as

L(𝒔h,𝝅v)=i=1kmin{α,ωi(𝝅v(𝒔h))}𝐿subscript𝒔subscript𝝅𝑣superscriptsubscript𝑖1𝑘𝛼subscript𝜔𝑖subscript𝝅𝑣subscript𝒔L(\bm{s}_{h},\bm{\pi}_{v})=\sum_{i=1}^{k}\min\{\alpha,\omega_{i}(\bm{\pi}_{v}(\bm{s}_{h}))\} (A.65)

where ωi(𝝅v(𝒔h))subscript𝜔𝑖subscript𝝅𝑣subscript𝒔\omega_{i}(\bm{\pi}_{v}(\bm{s}_{h})) is defined in (A.64). From hisubscript𝑖h_{i} in (A.62), we have

hi={1,i[g1]2,ig1[g2]nI,it=1nI1gt[gnI]subscript𝑖cases1𝑖delimited-[]subscript𝑔12𝑖subscript𝑔1delimited-[]subscript𝑔2otherwisesubscript𝑛𝐼𝑖superscriptsubscript𝑡1subscript𝑛𝐼1subscript𝑔𝑡delimited-[]subscript𝑔subscript𝑛𝐼h_{i}=\begin{cases}1,&i\in[g_{1}]\\ 2,&i-g_{1}\in[g_{2}]\\ &\vdots\\ n_{I},&i-\sum_{t=1}^{n_{I}-1}g_{t}\in[g_{n_{I}}]\end{cases} (A.66)

If we define

Im={i[k]:hi=m},superscriptsubscript𝐼𝑚conditional-set𝑖delimited-[]𝑘subscript𝑖𝑚I_{m}^{*}=\{i\in[k]:h_{i}=m\},

then L(𝒔h,𝝅v)𝐿subscript𝒔subscript𝝅𝑣L(\bm{s}_{h},\bm{\pi}_{v}) in (A.65) can be expressed as

L(𝒔h,𝝅v)=i=1kmin{α,(nIhi)βI+(nnIi+hi)βc}𝐿subscript𝒔subscript𝝅𝑣superscriptsubscript𝑖1𝑘𝛼subscript𝑛𝐼subscript𝑖subscript𝛽𝐼𝑛subscript𝑛𝐼𝑖subscript𝑖subscript𝛽𝑐\displaystyle L(\bm{s}_{h},\bm{\pi}_{v})=\sum_{i=1}^{k}\min\{\alpha,(n_{I}-h_{i})\beta_{I}+(n-n_{I}-i+h_{i})\beta_{c}\}
=m=1nIiImmin{α,(nIm)βI+(nnIi+m)βc}absentsuperscriptsubscript𝑚1subscript𝑛𝐼subscript𝑖superscriptsubscript𝐼𝑚𝛼subscript𝑛𝐼𝑚subscript𝛽𝐼𝑛subscript𝑛𝐼𝑖𝑚subscript𝛽𝑐\displaystyle=\sum_{m=1}^{n_{I}}\sum_{i\in I_{m}^{*}}\min\{\alpha,(n_{I}-m)\beta_{I}+(n-n_{I}-i+m)\beta_{c}\}
=m=1nIiImmin{α,ρmβI+(nρmi)βc}absentsuperscriptsubscript𝑚1subscript𝑛𝐼subscript𝑖superscriptsubscript𝐼𝑚𝛼subscript𝜌𝑚subscript𝛽𝐼𝑛subscript𝜌𝑚𝑖subscript𝛽𝑐\displaystyle=\sum_{m=1}^{n_{I}}\sum_{i\in I_{m}^{*}}\min\{\alpha,\rho_{m}\beta_{I}+(n-\rho_{m}-i)\beta_{c}\} (A.67)

where ρmsubscript𝜌𝑚\rho_{m} is defined in (6).

Using (A.66), we have

Im={t=1m1gt+1,,t=1m1gt+gm}.superscriptsubscript𝐼𝑚superscriptsubscript𝑡1𝑚1subscript𝑔𝑡1superscriptsubscript𝑡1𝑚1subscript𝑔𝑡subscript𝑔𝑚I_{m}^{*}=\{\sum_{t=1}^{m-1}g_{t}+1,\cdots,\sum_{t=1}^{m-1}g_{t}+g_{m}\}.

for m[nI]𝑚delimited-[]subscript𝑛𝐼m\in[n_{I}]. Therefore, iIm𝑖superscriptsubscript𝐼𝑚i\in I_{m}^{*} can be represented as

i=t=1m1gt+l𝑖superscriptsubscript𝑡1𝑚1subscript𝑔𝑡𝑙i=\sum_{t=1}^{m-1}g_{t}+l

for l[gm]𝑙delimited-[]subscript𝑔𝑚l\in[g_{m}]. Using this notation, (A) can be written as

L(𝒔h,𝝅v)=m=1nIl=1gmmin{α,ρmβI+(nρmsm(l))βc}𝐿subscript𝒔subscript𝝅𝑣superscriptsubscript𝑚1subscript𝑛𝐼superscriptsubscript𝑙1subscript𝑔𝑚𝛼subscript𝜌𝑚subscript𝛽𝐼𝑛subscript𝜌𝑚superscriptsubscript𝑠𝑚𝑙subscript𝛽𝑐\displaystyle L(\bm{s}_{h},\bm{\pi}_{v})=\sum_{m=1}^{n_{I}}\sum_{l=1}^{g_{m}}\min\{\alpha,\rho_{m}\beta_{I}+(n-\rho_{m}-s_{m}^{(l)})\beta_{c}\} (A.68)

where sm(l)=t=1m1gt+lsuperscriptsubscript𝑠𝑚𝑙superscriptsubscript𝑡1𝑚1subscript𝑔𝑡𝑙s_{m}^{(l)}=\sum_{t=1}^{m-1}g_{t}+l. This expression reduces to (A.1). This completes the proof of Part II-2. Therefore, the storage capacity of clustered DSS is as stated in Theorem 1.

Appendix B Proof of Theorem 3

We begin with introducing properties of the parameters ztsubscript𝑧𝑡z_{t} and htsubscript𝑡h_{t} defined in (25) and (26).

Proposition 4.
hksubscript𝑘\displaystyle h_{k} =nI,absentsubscript𝑛𝐼\displaystyle=n_{I}, (B.1)
zksubscript𝑧𝑘\displaystyle z_{k} =(nk)ϵ.absent𝑛𝑘italic-ϵ\displaystyle=(n-k)\epsilon. (B.2)
Proof.

Since we consider k>nI𝑘subscript𝑛𝐼k>n_{I} case as stated in (3), we have

gi1i[nI]formulae-sequencesubscript𝑔𝑖1for-all𝑖delimited-[]subscript𝑛𝐼g_{i}\geq 1\quad\forall i\in[n_{I}]

for {gi}i=1nIsuperscriptsubscriptsubscript𝑔𝑖𝑖1subscript𝑛𝐼\{g_{i}\}_{i=1}^{n_{I}} defined in (7). Combining with (9) and (26), we can conclude that hk=nIsubscript𝑘subscript𝑛𝐼h_{k}=n_{I}. Finally, zk=(nk)ϵsubscript𝑧𝑘𝑛𝑘italic-ϵz_{k}=(n-k)\epsilon is from (25) and (B.1). ∎

First, consider the ϵ1nkitalic-ϵ1𝑛𝑘\epsilon\geq\frac{1}{n-k} case. From (20), data \mathcal{M} can be reliably stored with node storage α=/k𝛼𝑘\alpha=\mathcal{M}/k if the repair bandwidth satisfies γγ𝛾superscript𝛾\gamma\geq\gamma^{*}, where

γsuperscript𝛾\displaystyle\gamma^{*} =(k1)/ksk1=k1sk1absent𝑘1𝑘subscript𝑠𝑘1𝑘1subscript𝑠𝑘1\displaystyle=\frac{\mathcal{M}-(k-1)\mathcal{M}/k}{s_{k-1}}=\frac{\mathcal{M}}{k}\frac{1}{s_{k-1}}
=k(nk)ϵ(nI1)+(nnI)ϵabsent𝑘𝑛𝑘italic-ϵsubscript𝑛𝐼1𝑛subscript𝑛𝐼italic-ϵ\displaystyle=\frac{\mathcal{M}}{k}\frac{(n-k)\epsilon}{(n_{I}-1)+(n-n_{I})\epsilon}

where the last equality is from (23) and (B.2). Thus, α=/k𝛼𝑘\alpha=\mathcal{M}/k is achievable with finite γ𝛾\gamma, when ϵ1nkitalic-ϵ1𝑛𝑘\epsilon\geq\frac{1}{n-k}.

Second, we prove that it is impossible to achieve α=/k𝛼𝑘\alpha=\mathcal{M}/k for 0ϵ<1nk0italic-ϵ1𝑛𝑘0\leq\epsilon<\frac{1}{n-k}, in order to reliably store file \mathcal{M}. Recall that the minimum storage for 0ϵ<1nk0italic-ϵ1𝑛𝑘0\leq\epsilon<\frac{1}{n-k} is

α=Mτ+i=τ+1kzi𝛼𝑀𝜏superscriptsubscript𝑖𝜏1𝑘subscript𝑧𝑖\alpha=\frac{M}{\tau+\sum_{i=\tau+1}^{k}z_{i}} (B.3)

from (28). From (22) and (B.2), we have zi<1subscript𝑧𝑖1z_{i}<1 for i=τ+1,τ+2,,k𝑖𝜏1𝜏2𝑘i=\tau+1,\tau+2,\cdots,k. Therefore,

τ+i=τ+1kzi<τ+(k(τ+1)+1)=k𝜏superscriptsubscript𝑖𝜏1𝑘subscript𝑧𝑖𝜏𝑘𝜏11𝑘\tau+\sum_{i=\tau+1}^{k}z_{i}<\tau+(k-(\tau+1)+1)=k

holds, which result in

Mτ+i=τ+1kzi>k.𝑀𝜏superscriptsubscript𝑖𝜏1𝑘subscript𝑧𝑖𝑘\frac{M}{\tau+\sum_{i=\tau+1}^{k}z_{i}}>\frac{\mathcal{M}}{k}. (B.4)

Thus, the 0ϵ1nk0italic-ϵ1𝑛𝑘0\leq\epsilon\leq\frac{1}{n-k} case has the minimum storage α𝛼\alpha greater than /k𝑘\mathcal{M}/k, which completes the proof of Theorem 3.

Appendix C Proof of Theorem 4

First, we prove (30). To begin with, we obtain the expression of αmsr(ϵ)superscriptsubscript𝛼𝑚𝑠𝑟italic-ϵ\alpha_{msr}^{(\epsilon)}, for ϵ=0,1italic-ϵ01\epsilon=0,1. From (28), we obtain

αmsr(0)superscriptsubscript𝛼𝑚𝑠𝑟0\displaystyle\alpha_{msr}^{(0)} =τ+i=τ+1kzi,absent𝜏superscriptsubscript𝑖𝜏1𝑘subscript𝑧𝑖\displaystyle=\frac{\mathcal{M}}{\tau+\sum_{i=\tau+1}^{k}z_{i}}, (C.1)
αmsr(1)superscriptsubscript𝛼𝑚𝑠𝑟1\displaystyle\alpha_{msr}^{(1)} =k.absent𝑘\displaystyle=\frac{\mathcal{M}}{k}.

We further simplify the expression for αmsr(0)superscriptsubscript𝛼𝑚𝑠𝑟0\alpha_{msr}^{(0)} as follows. Recall

zt=nIhtsubscript𝑧𝑡subscript𝑛𝐼subscript𝑡z_{t}=n_{I}-h_{t} (C.2)

for t[k]𝑡delimited-[]𝑘t\in[k] from (25), when ϵ=0italic-ϵ0\epsilon=0 holds. Note that we have

zt={0,tkknI+11,t=kknIsubscript𝑧𝑡cases0𝑡𝑘𝑘subscript𝑛𝐼11𝑡𝑘𝑘subscript𝑛𝐼z_{t}=\begin{cases}0,&t\geq k-\lfloor\frac{k}{n_{I}}\rfloor+1\\ 1,&t=k-\lfloor\frac{k}{n_{I}}\rfloor\end{cases} (C.3)

from the following reason. First, from (26) and (C.2), zt=0subscript𝑧𝑡0z_{t}=0 holds for

t𝑡\displaystyle t l=1nI1gl+1=l=1nIglgnI+1=kknI+1absentsuperscriptsubscript𝑙1subscript𝑛𝐼1subscript𝑔𝑙1superscriptsubscript𝑙1subscript𝑛𝐼subscript𝑔𝑙subscript𝑔subscript𝑛𝐼1𝑘𝑘subscript𝑛𝐼1\displaystyle\geq\sum_{l=1}^{n_{I}-1}g_{l}+1=\sum_{l=1}^{n_{I}}g_{l}-g_{n_{I}}+1=k-\lfloor\frac{k}{n_{I}}\rfloor+1

where the last equality is from (9) and (7). Similarly, we can prove that zt=1subscript𝑧𝑡1z_{t}=1 holds for t=kknI𝑡𝑘𝑘subscript𝑛𝐼t=k-\lfloor\frac{k}{n_{I}}\rfloor. From (C.3) and (22), we obtain

τ=kknI𝜏𝑘𝑘subscript𝑛𝐼\tau=k-\lfloor\frac{k}{n_{I}}\rfloor (C.4)

when ϵ=0italic-ϵ0\epsilon=0. Combining (C.1), (C.3) and (C.4), we have

αmsr(0)=kknI.superscriptsubscript𝛼𝑚𝑠𝑟0𝑘𝑘subscript𝑛𝐼\alpha_{msr}^{(0)}=\frac{\mathcal{M}}{k-\lfloor\frac{k}{n_{I}}\rfloor}. (C.5)

Then, using R=k/n𝑅𝑘𝑛R=k/n and σ=L2/n𝜎superscript𝐿2𝑛\sigma=L^{2}/n,

αmsr(0)αmsr(1)superscriptsubscript𝛼𝑚𝑠𝑟0superscriptsubscript𝛼𝑚𝑠𝑟1\displaystyle\frac{\alpha_{msr}^{(0)}}{\alpha_{msr}^{(1)}} =kkknI=nRnRRLabsent𝑘𝑘𝑘subscript𝑛𝐼𝑛𝑅𝑛𝑅𝑅𝐿\displaystyle=\frac{k}{k-\lfloor\frac{k}{n_{I}}\rfloor}=\frac{nR}{nR-\lfloor RL\rfloor}
=nRnRRnσ=RRRnσ/n.absent𝑛𝑅𝑛𝑅𝑅𝑛𝜎𝑅𝑅𝑅𝑛𝜎𝑛\displaystyle=\frac{nR}{nR-\lfloor R\sqrt{n\sigma}\rfloor}=\frac{R}{R-\lfloor R\sqrt{n\sigma}\rfloor/n}.

Thus, for arbitrary fixed R𝑅R and σ𝜎\sigma,

limnαmsr(0)αmsr(1)=1.subscript𝑛superscriptsubscript𝛼𝑚𝑠𝑟0superscriptsubscript𝛼𝑚𝑠𝑟11\lim\limits_{n\rightarrow\infty}\frac{\alpha_{msr}^{(0)}}{\alpha_{msr}^{(1)}}=1.

Therefore, αmsr(0)superscriptsubscript𝛼𝑚𝑠𝑟0\alpha_{msr}^{(0)} is asymptotically equivalent to αmsr(1)superscriptsubscript𝛼𝑚𝑠𝑟1\alpha_{msr}^{(1)}.

Second, we prove (31). Note that from (29), αmbr(ϵ)=γmbr(ϵ)superscriptsubscript𝛼𝑚𝑏𝑟italic-ϵsuperscriptsubscript𝛾𝑚𝑏𝑟italic-ϵ\alpha_{mbr}^{(\epsilon)}=\gamma_{mbr}^{(\epsilon)} holds for arbitrary 0ϵ10italic-ϵ10\leq\epsilon\leq 1. Therefore, all we need to prove is

γmbr(0)γmbr(1).superscriptsubscript𝛾𝑚𝑏𝑟0superscriptsubscript𝛾𝑚𝑏𝑟1\gamma_{mbr}^{(0)}\rightarrow\gamma_{mbr}^{(1)}.

To begin with, we obtain the expression for γmbr(ϵ)superscriptsubscript𝛾𝑚𝑏𝑟italic-ϵ\gamma_{mbr}^{(\epsilon)}, when ϵ=0,1italic-ϵ01\epsilon=0,1. For ϵ=1italic-ϵ1\epsilon=1, ztsubscript𝑧𝑡z_{t} in (25) is

zt=ntsubscript𝑧𝑡𝑛𝑡z_{t}=n-t (C.6)

for t[k]𝑡delimited-[]𝑘t\in[k]. Moreover, from (23),

s0=i=1kzin1subscript𝑠0superscriptsubscript𝑖1𝑘subscript𝑧𝑖𝑛1s_{0}=\frac{\sum_{i=1}^{k}z_{i}}{n-1} (C.7)

for ϵ=1italic-ϵ1\epsilon=1. Therefore, from (29), (C.6) and (C.7),

γmbr(1)=s0superscriptsubscript𝛾𝑚𝑏𝑟1subscript𝑠0\displaystyle\gamma_{mbr}^{(1)}=\frac{\mathcal{M}}{s_{0}} =(n1)i=1k(ni)=k2(n1)2nk1.absent𝑛1superscriptsubscript𝑖1𝑘𝑛𝑖𝑘2𝑛12𝑛𝑘1\displaystyle=\frac{(n-1)\mathcal{M}}{\sum_{i=1}^{k}(n-i)}=\frac{\mathcal{M}}{k}\frac{2(n-1)}{2n-k-1}. (C.8)

Now we focus on the case of ϵ=0italic-ϵ0\epsilon=0. First, let q𝑞q and r𝑟r be

q𝑞\displaystyle q knI,absent𝑘subscript𝑛𝐼\displaystyle\coloneqq\lfloor\frac{k}{n_{I}}\rfloor, (C.9)
r𝑟\displaystyle r (kmodnI),absent𝑘modsubscript𝑛𝐼\displaystyle\coloneqq(k\ \mathrm{mod}\ n_{I}), (C.10)

which represent the quotient and remainder of k/nI𝑘subscript𝑛𝐼k/n_{I}. Note that

qnI+r=k.𝑞subscript𝑛𝐼𝑟𝑘qn_{I}+r=k. (C.11)

Then, we have

t=1nItgtsuperscriptsubscript𝑡1subscript𝑛𝐼𝑡subscript𝑔𝑡\displaystyle\sum_{t=1}^{n_{I}}tg_{t} =t=1r(q+1)t+t=r+1nIqt=qt=1nIt+t=1rtabsentsuperscriptsubscript𝑡1𝑟𝑞1𝑡superscriptsubscript𝑡𝑟1subscript𝑛𝐼𝑞𝑡𝑞superscriptsubscript𝑡1subscript𝑛𝐼𝑡superscriptsubscript𝑡1𝑟𝑡\displaystyle=\sum_{t=1}^{r}(q+1)t+\sum_{t=r+1}^{n_{I}}qt=q\sum_{t=1}^{n_{I}}t+\sum_{t=1}^{r}t
=qnI(nI+1)2+r(r+1)2=12(qnI2+r2+k)absent𝑞subscript𝑛𝐼subscript𝑛𝐼12𝑟𝑟1212𝑞superscriptsubscript𝑛𝐼2superscript𝑟2𝑘\displaystyle=q\frac{n_{I}(n_{I}+1)}{2}+\frac{r(r+1)}{2}=\frac{1}{2}(qn_{I}^{2}+r^{2}+k) (C.12)

where the last equality is from (C.11). From (C.2) and (A.66), we have

i=1kzisuperscriptsubscript𝑖1𝑘subscript𝑧𝑖\displaystyle\sum_{i=1}^{k}z_{i} =t=1nI(nIt)gt=nIk12(qnI2+r2+k)absentsuperscriptsubscript𝑡1subscript𝑛𝐼subscript𝑛𝐼𝑡subscript𝑔𝑡subscript𝑛𝐼𝑘12𝑞superscriptsubscript𝑛𝐼2superscript𝑟2𝑘\displaystyle=\sum_{t=1}^{n_{I}}(n_{I}-t)g_{t}=n_{I}k-\frac{1}{2}(qn_{I}^{2}+r^{2}+k)
=(nI1)k12(qnI2+r2k)absentsubscript𝑛𝐼1𝑘12𝑞superscriptsubscript𝑛𝐼2superscript𝑟2𝑘\displaystyle=(n_{I}-1)k-\frac{1}{2}(qn_{I}^{2}+r^{2}-k) (C.13)

where the second last equality is from (9) and (C). Moreover, using (C.11), we have

qnI2+r2k𝑞superscriptsubscript𝑛𝐼2superscript𝑟2𝑘\displaystyle qn_{I}^{2}+r^{2}-k =qnI2+r2qnIrabsent𝑞superscriptsubscript𝑛𝐼2superscript𝑟2𝑞subscript𝑛𝐼𝑟\displaystyle=qn_{I}^{2}+r^{2}-qn_{I}-r
=(qnI+r)(nI1)r(nIr)absent𝑞subscript𝑛𝐼𝑟subscript𝑛𝐼1𝑟subscript𝑛𝐼𝑟\displaystyle=(qn_{I}+r)(n_{I}-1)-r(n_{I}-r)
(qnI+r)(nI1)=k(nI1)absent𝑞subscript𝑛𝐼𝑟subscript𝑛𝐼1𝑘subscript𝑛𝐼1\displaystyle\leq(qn_{I}+r)(n_{I}-1)=k(n_{I}-1) (C.14)

where the equality holds if and only if r=0𝑟0r=0. Furthermore,

s0=i=1kzinI1subscript𝑠0superscriptsubscript𝑖1𝑘subscript𝑧𝑖subscript𝑛𝐼1s_{0}=\frac{\sum_{i=1}^{k}z_{i}}{n_{I}-1} (C.15)

for ϵ=0italic-ϵ0\epsilon=0 from (23). Combining (29), (C), (C) and (C.15) result in

γmbr(0)=s0superscriptsubscript𝛾𝑚𝑏𝑟0subscript𝑠0\displaystyle\gamma_{mbr}^{(0)}=\frac{\mathcal{M}}{s_{0}} 2k.absent2𝑘\displaystyle\leq\frac{2\mathcal{M}}{k}. (C.16)

From (C.8) and (C.16), we have

γmbr(0)γmbr(1)superscriptsubscript𝛾𝑚𝑏𝑟0superscriptsubscript𝛾𝑚𝑏𝑟1\displaystyle\gamma_{mbr}^{(0)}-\gamma_{mbr}^{(1)} k(22(n1)2nk1)=k2(nk)2nk1absent𝑘22𝑛12𝑛𝑘1𝑘2𝑛𝑘2𝑛𝑘1\displaystyle\leq\frac{\mathcal{M}}{k}\left(2-\frac{2(n-1)}{2n-k-1}\right)=\frac{\mathcal{M}}{k}\frac{2(n-k)}{2n-k-1}
=nR2n(1R)n(2R)1absent𝑛𝑅2𝑛1𝑅𝑛2𝑅1\displaystyle=\frac{\mathcal{M}}{nR}\frac{2n(1-R)}{n(2-R)-1}

where R=k/n𝑅𝑘𝑛R=k/n. Thus, for arbitrary n𝑛n,

γmbr(0)γmbr(1)superscriptsubscript𝛾𝑚𝑏𝑟0superscriptsubscript𝛾𝑚𝑏𝑟1\gamma_{mbr}^{(0)}\rightarrow\gamma_{mbr}^{(1)}

as R1𝑅1R\rightarrow 1. This completes the proof of (31). Finally, (C.8) and (C.16) provides

γmbr(0)γmbr(1)superscriptsubscript𝛾𝑚𝑏𝑟0superscriptsubscript𝛾𝑚𝑏𝑟1\displaystyle\frac{\gamma_{mbr}^{(0)}}{\gamma_{mbr}^{(1)}} 2nk1n1=2k1n1,absent2𝑛𝑘1𝑛12𝑘1𝑛1\displaystyle\leq\frac{2n-k-1}{n-1}=2-\frac{k-1}{n-1},

which completes the proof of (32).

Appendix D Proof of Theorem 5

Recall the definition of an (n,l0,m0,,α)𝑛subscript𝑙0subscript𝑚0𝛼(n,l_{0},m_{0},\mathcal{M},\alpha)-LRC which appear right before Theorem 5. Moreover, recall that the repair locality of a code is defined as the number of nodes to be contacted in the node repair process [32]. Since each cluster contains nIsubscript𝑛𝐼n_{I} nodes, every node in a DSS with ϵ=0italic-ϵ0\epsilon=0 has the repair locality of

l0=nI1.subscript𝑙0subscript𝑛𝐼1l_{0}=n_{I}-1. (D.1)

Moreover, note that for any code with minimum distance m𝑚m, the original file \mathcal{M} can be retrieved by contacting nm+1𝑛𝑚1n-m+1 coded symbols [38]. Since the present paper considers DSSs such that contacting any k𝑘k nodes can retrieve the original file \mathcal{M}, we have the minimum distance of

m0=nk+1.subscript𝑚0𝑛𝑘1m_{0}=n-k+1. (D.2)

Thus, the intra-cluster repairable code defined in Section IV-C is a (n,l0,m0,,α)𝑛subscript𝑙0subscript𝑚0𝛼(n,l_{0},m_{0},\mathcal{M},\alpha)-LRC.

Now we show that (34) holds. Note that from Fig. 10, we obtain ααmsr(0)𝛼superscriptsubscript𝛼𝑚𝑠𝑟0\alpha\geq\alpha_{msr}^{(0)} for ϵ=0italic-ϵ0\epsilon=0, where

αmsr(0)=kknIsuperscriptsubscript𝛼𝑚𝑠𝑟0𝑘𝑘subscript𝑛𝐼\alpha_{msr}^{(0)}=\frac{\mathcal{M}}{k-\lfloor\frac{k}{n_{I}}\rfloor} (D.3)

holds according to (C.1). Thus, (34) is proven by showing

m0nαmsr(0)l0αmsr(0)+2.subscript𝑚0𝑛superscriptsubscript𝛼𝑚𝑠𝑟0subscript𝑙0superscriptsubscript𝛼𝑚𝑠𝑟02m_{0}\leq n-\bigg{\lceil}\frac{\mathcal{M}}{\alpha_{msr}^{(0)}}\bigg{\rceil}-\bigg{\lceil}\frac{\mathcal{M}}{l_{0}\alpha_{msr}^{(0)}}\bigg{\rceil}+2. (D.4)

By plugging (D.1), (D.2) and (D.3) into (D.4), we have

nk+1𝑛𝑘1\displaystyle n-k+1 nkknIkknInI1+2absent𝑛𝑘𝑘subscript𝑛𝐼𝑘𝑘subscript𝑛𝐼subscript𝑛𝐼12\displaystyle\leq n-\bigg{\lceil}k-\bigg{\lfloor}\frac{k}{n_{I}}\bigg{\rfloor}\bigg{\rceil}-\bigg{\lceil}\frac{k-\lfloor\frac{k}{n_{I}}\rfloor}{n_{I}-1}\bigg{\rceil}+2
=nk+knIkknInI1+2.absent𝑛𝑘𝑘subscript𝑛𝐼𝑘𝑘subscript𝑛𝐼subscript𝑛𝐼12\displaystyle=n-k+\bigg{\lfloor}\frac{k}{n_{I}}\bigg{\rfloor}-\bigg{\lceil}\frac{k-\lfloor\frac{k}{n_{I}}\rfloor}{n_{I}-1}\bigg{\rceil}+2.

Therefore, all we need to prove is

0knIkknInI1+1,0𝑘subscript𝑛𝐼𝑘𝑘subscript𝑛𝐼subscript𝑛𝐼110\leq\bigg{\lfloor}\frac{k}{n_{I}}\bigg{\rfloor}-\bigg{\lceil}\frac{k-\lfloor\frac{k}{n_{I}}\rfloor}{n_{I}-1}\bigg{\rceil}+1, (D.5)

which is proved as follows. Using q𝑞q and r𝑟r defined in (C.9) and (C.10), the right-hand-side (RHS) of (D.5) is

RHS𝑅𝐻𝑆\displaystyle RHS =qqnI+rqnI1+1absent𝑞𝑞subscript𝑛𝐼𝑟𝑞subscript𝑛𝐼11\displaystyle=q-\bigg{\lceil}\frac{qn_{I}+r-q}{n_{I}-1}\bigg{\rceil}+1
={q(q+1)+1=0,r0qq+1=1,r=0absentcases𝑞𝑞110𝑟0𝑞𝑞11𝑟0\displaystyle=\begin{cases}q-(q+1)+1=0,&\quad r\neq 0\\ q-q+1=1,&\quad r=0\\ \end{cases}

Thus, (D.5) holds, where the equality condition is r0𝑟0r\neq 0, or equivalently (kmodnI)0𝑘modsubscript𝑛𝐼0(k\ \mathrm{mod}\ n_{I})\neq 0. Therefore, (34) holds, where the equality condition is α=αmsr(0)𝛼superscriptsubscript𝛼𝑚𝑠𝑟0\alpha=\alpha_{msr}^{(0)} and (kmodnI)0𝑘modsubscript𝑛𝐼0(k\ \mathrm{mod}\ n_{I})\neq 0. This completes the proof of Theorem 5.

Appendix E Proof of Theorem 6

According to (A.20) and (A.68) in Appendix B, the capacity can be expressed as

𝒞=i=1kmin{α,ωi(𝝅v(𝒔h))}.𝒞superscriptsubscript𝑖1𝑘𝛼subscript𝜔𝑖subscript𝝅𝑣subscript𝒔\mathcal{C}=\sum_{i=1}^{k}\min\{\alpha,\omega_{i}(\bm{\pi}_{v}(\bm{s}_{h}))\}. (E.1)

Using (A.64) and (A.66), ωi(𝝅v(𝒔h))subscript𝜔𝑖subscript𝝅𝑣subscript𝒔\omega_{i}(\bm{\pi}_{v}(\bm{s}_{h})) in (E.1), or simply ωisubscript𝜔𝑖\omega_{i} has the following property:

ωi+1={ωiβI,iIGωiβc,i[k]IGsubscript𝜔𝑖1casessubscript𝜔𝑖subscript𝛽𝐼𝑖subscript𝐼𝐺subscript𝜔𝑖subscript𝛽𝑐𝑖delimited-[]𝑘subscript𝐼𝐺\omega_{i+1}=\begin{cases}\omega_{i}-\beta_{I},&i\in I_{G}\\ \omega_{i}-\beta_{c},&i\in[k]\setminus I_{G}\end{cases} (E.2)

where IG={gm}m=1nI1.subscript𝐼𝐺superscriptsubscriptsubscript𝑔𝑚𝑚1subscript𝑛𝐼1I_{G}=\{g_{m}\}_{m=1}^{n_{I}-1}. Note that gnI=knIsubscript𝑔subscript𝑛𝐼𝑘subscript𝑛𝐼g_{n_{I}}=\lfloor\frac{k}{n_{I}}\rfloor from (7). Therefore, k0subscript𝑘0k_{0} in (37) can be expressed as

k0=kknI=kgnIk1subscript𝑘0𝑘𝑘subscript𝑛𝐼𝑘subscript𝑔subscript𝑛𝐼𝑘1k_{0}=k-\lfloor\frac{k}{n_{I}}\rfloor=k-g_{n_{I}}\leq k-1 (E.3)

where the last inequality is from (3). Combining (26) and (E.3) result in

hk0=nI1.subscriptsubscript𝑘0subscript𝑛𝐼1h_{k_{0}}=n_{I}-1. (E.4)

From (A.64), (E.3) and (E.4), we have

ωk0subscript𝜔subscript𝑘0\displaystyle\omega_{k_{0}} =(nIhk0)βI+(nk0nI+hk0)βcabsentsubscript𝑛𝐼subscriptsubscript𝑘0subscript𝛽𝐼𝑛subscript𝑘0subscript𝑛𝐼subscriptsubscript𝑘0subscript𝛽𝑐\displaystyle=(n_{I}-h_{k_{0}})\beta_{I}+(n-k_{0}-n_{I}+h_{k_{0}})\beta_{c}
=βI+(nk01)βcβI+(nk)βcβI=α.absentsubscript𝛽𝐼𝑛subscript𝑘01subscript𝛽𝑐subscript𝛽𝐼𝑛𝑘subscript𝛽𝑐subscript𝛽𝐼𝛼\displaystyle=\beta_{I}+(n-k_{0}-1)\beta_{c}\geq\beta_{I}+(n-k)\beta_{c}\geq\beta_{I}=\alpha. (E.5)

where the last equality holds due to the assumption of βI=αsubscript𝛽𝐼𝛼\beta_{I}=\alpha in the setting of Theorem 6. Since (ωi)i=1ksuperscriptsubscriptsubscript𝜔𝑖𝑖1𝑘(\omega_{i})_{i=1}^{k} is a decreasing sequence from (E.2), the result of (E) implies that

ωiα,1ik0.formulae-sequencesubscript𝜔𝑖𝛼1𝑖subscript𝑘0\omega_{i}\geq\alpha,\quad\quad 1\leq i\leq k_{0}. (E.6)

Thus, the capacity expression in (E.1) can be expressed as

𝒞𝒞\displaystyle\mathcal{C} ={k0α+i=k0+1kωi,ωk0+1αmα+i=m+1kωi,ωm+1α<ωm(k0+1mk1)kα,α<ωkabsentcasessubscript𝑘0𝛼superscriptsubscript𝑖subscript𝑘01𝑘subscript𝜔𝑖subscript𝜔subscript𝑘01𝛼𝑚𝛼superscriptsubscript𝑖𝑚1𝑘subscript𝜔𝑖subscript𝜔𝑚1𝛼subscript𝜔𝑚otherwisesubscript𝑘01𝑚𝑘1𝑘𝛼𝛼subscript𝜔𝑘\displaystyle=\begin{cases}k_{0}\alpha+\sum_{i=k_{0}+1}^{k}\omega_{i},&\omega_{k_{0}+1}\leq\alpha\\ m\alpha+\sum_{i=m+1}^{k}\omega_{i},&\omega_{m+1}\leq\alpha<\omega_{m}\\ &\ \ \ (k_{0}+1\leq m\leq k-1)\\ k\alpha,&\alpha<\omega_{k}\end{cases} (E.7)

Note that from (A.64) and (A.66), we have ωi=(ni)βcsubscript𝜔𝑖𝑛𝑖subscript𝛽𝑐\omega_{i}=(n-i)\beta_{c} for i=k0+1,k0+2,,k𝑖subscript𝑘01subscript𝑘02𝑘i=k_{0}+1,k_{0}+2,\cdots,k. Therefore, 𝒞𝒞\mathcal{C} in (E.7) is

𝒞𝒞\displaystyle\mathcal{C} ={k0α+i=k0+1k(ni)βc,0βcαnk01mα+i=m+1k(ni)βc,αnm<βcαnm1(k0+1mk1)kα,αnk<βc,absentcasessubscript𝑘0𝛼superscriptsubscript𝑖subscript𝑘01𝑘𝑛𝑖subscript𝛽𝑐0subscript𝛽𝑐𝛼𝑛subscript𝑘01𝑚𝛼superscriptsubscript𝑖𝑚1𝑘𝑛𝑖subscript𝛽𝑐𝛼𝑛𝑚subscript𝛽𝑐𝛼𝑛𝑚1otherwisesubscript𝑘01𝑚𝑘1𝑘𝛼𝛼𝑛𝑘subscript𝛽𝑐\displaystyle=\begin{cases}k_{0}\alpha+\sum_{i=k_{0}+1}^{k}(n-i)\beta_{c},&0\leq\beta_{c}\leq\frac{\alpha}{n-k_{0}-1}\\ m\alpha+\sum_{i=m+1}^{k}(n-i)\beta_{c},&\frac{\alpha}{n-m}<\beta_{c}\leq\frac{\alpha}{n-m-1}\\ &\ \ \ (k_{0}+1\leq m\leq k-1)\\ k\alpha,&\frac{\alpha}{n-k}<\beta_{c},\end{cases} (E.8)

which is illustrated as a piecewise linear function of βcsubscript𝛽𝑐\beta_{c} in Fig. 28. Based on (E.8), the sequence (Tm)m=k0ksuperscriptsubscriptsubscript𝑇𝑚𝑚subscript𝑘0𝑘(T_{m})_{m=k_{0}}^{k} in this figure has the following expression:

Tm={k0α,m=k0(m+i=m+1k(ni)nm)α,k0+1mk1kα,m=ksubscript𝑇𝑚casessubscript𝑘0𝛼𝑚subscript𝑘0𝑚superscriptsubscript𝑖𝑚1𝑘𝑛𝑖𝑛𝑚𝛼subscript𝑘01𝑚𝑘1𝑘𝛼𝑚𝑘T_{m}=\begin{cases}k_{0}\alpha,&m=k_{0}\\ (m+\frac{\sum_{i=m+1}^{k}(n-i)}{n-m})\alpha,&k_{0}+1\leq m\leq k-1\\ k\alpha,&m=k\end{cases} (E.9)
Refer to caption
Figure 28: Capacity as a function of βcsubscript𝛽𝑐\beta_{c}

From Fig. 28, we can conclude that 𝒞𝒞\mathcal{C}\geq\mathcal{M} holds if and only if βcβcsubscript𝛽𝑐superscriptsubscript𝛽𝑐\beta_{c}\geq\beta_{c}^{*} where

βc={0,[0,Tk0]mαi=m+1k(ni),(Tm,Tm+1](m=k0,k0+1,,k1),(Tk,).superscriptsubscript𝛽𝑐cases00subscript𝑇subscript𝑘0𝑚𝛼superscriptsubscript𝑖𝑚1𝑘𝑛𝑖subscript𝑇𝑚subscript𝑇𝑚1otherwise𝑚subscript𝑘0subscript𝑘01𝑘1subscript𝑇𝑘\beta_{c}^{*}=\begin{cases}0,&\mathcal{M}\in[0,T_{k_{0}}]\\ \frac{\mathcal{M}-m\alpha}{\sum_{i=m+1}^{k}(n-i)},&\mathcal{M}\in(T_{m},T_{m+1}]\\ &\quad(m=k_{0},k_{0}+1,\cdots,k-1)\\ \infty,&\mathcal{M}\in(T_{k},\infty).\end{cases} (E.10)

Using Tmsubscript𝑇𝑚T_{m} in (E.9) and fmsubscript𝑓𝑚f_{m} in (36), (E.10) reduces to (35), which completes the proof.

Appendix F Proofs of Corollaries

F-A Proof of Corollary 1

From the proof of Theorem 1, the capacity expression is equal to (A), which is

𝒞=i=1kmin{α,(nIhi)βI+(nnIi+hi)βc}.𝒞superscriptsubscript𝑖1𝑘𝛼subscript𝑛𝐼subscript𝑖subscript𝛽𝐼𝑛subscript𝑛𝐼𝑖subscript𝑖subscript𝛽𝑐\mathcal{C}=\sum_{i=1}^{k}\min\{\alpha,(n_{I}-h_{i})\beta_{I}+(n-n_{I}-i+h_{i})\beta_{c}\}.

where hisubscript𝑖h_{i} is defined in (A.60). Using ϵ=βc/βIitalic-ϵsubscript𝛽𝑐subscript𝛽𝐼\epsilon=\beta_{c}/\beta_{I} and (2), this can be rewritten as

𝒞=i=1kmin{α,(nnIi+hi)ϵ+(nIhi)(nnI)ϵ+(nI1)γ}.𝒞superscriptsubscript𝑖1𝑘𝛼𝑛subscript𝑛𝐼𝑖subscript𝑖italic-ϵsubscript𝑛𝐼subscript𝑖𝑛subscript𝑛𝐼italic-ϵsubscript𝑛𝐼1𝛾\mathcal{C}=\sum_{i=1}^{k}\min\{\alpha,\frac{(n-n_{I}-i+h_{i})\epsilon+(n_{I}-h_{i})}{(n-n_{I})\epsilon+(n_{I}-1)}\gamma\}. (F.1)

Using {zt}subscript𝑧𝑡\{z_{t}\} and {yt}subscript𝑦𝑡\{y_{t}\} defined in (25) and (24), the capacity expression reduces to

𝒞(α,γ)=t=1kmin{α,γyt},𝒞𝛼𝛾superscriptsubscript𝑡1𝑘𝛼𝛾subscript𝑦𝑡\mathcal{C}(\alpha,\gamma)=\sum_{t=1}^{k}\min\{\alpha,\frac{\gamma}{y_{t}}\}, (F.2)

which is a continuous function of γ𝛾\gamma.

Remark 1.

{zt}subscript𝑧𝑡\{z_{t}\} in (25) is a decreasing sequence of t𝑡t. Moreover, {yt}subscript𝑦𝑡\{y_{t}\} in (24) is an increasing sequence.

Proof.

Note that from (A.62),

ht+1={ht+1,tTht,t[k1]Tsubscript𝑡1casessubscript𝑡1𝑡𝑇subscript𝑡𝑡delimited-[]𝑘1𝑇h_{t+1}=\begin{cases}h_{t}+1,&t\in T\\ h_{t},&t\in[k-1]\setminus T\end{cases}

where T={g1,g1+g2,,m=1nI1gm}.𝑇subscript𝑔1subscript𝑔1subscript𝑔2superscriptsubscript𝑚1subscript𝑛𝐼1subscript𝑔𝑚T=\{g_{1},g_{1}+g_{2},\cdots,\sum_{m=1}^{n_{I}-1}g_{m}\}. Therefore, {zt}subscript𝑧𝑡\{z_{t}\} in (25) is a decreasing function of t𝑡t, which implies that {yt}subscript𝑦𝑡\{y_{t}\} is an increasing sequence. ∎

Moreover, note that βIαsubscript𝛽𝐼𝛼\beta_{I}\leq\alpha holds from the definition of βIsubscript𝛽𝐼\beta_{I} and α𝛼\alpha in Table I. Thus, combined with ϵ=βI/βcitalic-ϵsubscript𝛽𝐼subscript𝛽𝑐\epsilon=\beta_{I}/\beta_{c}, it is shown that γ𝛾\gamma in (2) is lower-bounded as

γ𝛾\displaystyle\gamma =(nI1)βI+(nnI)βcabsentsubscript𝑛𝐼1subscript𝛽𝐼𝑛subscript𝑛𝐼subscript𝛽𝑐\displaystyle=(n_{I}-1)\beta_{I}+(n-n_{I})\beta_{c}
={nI1+(nnI)ϵ}βI{nI1+(nnI)ϵ}α.absentsubscript𝑛𝐼1𝑛subscript𝑛𝐼italic-ϵsubscript𝛽𝐼subscript𝑛𝐼1𝑛subscript𝑛𝐼italic-ϵ𝛼\displaystyle=\{n_{I}-1+(n-n_{I})\epsilon\}\beta_{I}\leq\{n_{I}-1+(n-n_{I})\epsilon\}\alpha.

Here, we define

γ¯={nI1+(nnI)ϵ}α.¯𝛾subscript𝑛𝐼1𝑛subscript𝑛𝐼italic-ϵ𝛼\mkern 1.5mu\overline{\mkern-1.5mu\gamma\mkern-1.5mu}\mkern 1.5mu=\{n_{I}-1+(n-n_{I})\epsilon\}\alpha.

Then, the valid region of γ𝛾\gamma is expressed as γγ¯,𝛾¯𝛾\gamma\leq\mkern 1.5mu\overline{\mkern-1.5mu\gamma\mkern-1.5mu}\mkern 1.5mu, as illustrated in Figs. 29 and 30. The rest of the proof depends on the range of ϵitalic-ϵ\epsilon values; we first consider the 1nkϵ11𝑛𝑘italic-ϵ1\frac{1}{n-k}\leq\epsilon\leq 1 case, and then consider the 0ϵ<1nk0italic-ϵ1𝑛𝑘0\leq\epsilon<\frac{1}{n-k} case.

F-A1 If 1nkϵ11𝑛𝑘italic-ϵ1\frac{1}{n-k}\leq\epsilon\leq 1

Using (B.2), zk=(nk)ϵ1subscript𝑧𝑘𝑛𝑘italic-ϵ1z_{k}=(n-k)\epsilon\geq 1 holds. Combining with (24), we have yknI1+ϵ(nnI),subscript𝑦𝑘subscript𝑛𝐼1italic-ϵ𝑛subscript𝑛𝐼y_{k}\leq n_{I}-1+\epsilon(n-n_{I}), or equivalently, ykαγ¯.subscript𝑦𝑘𝛼¯𝛾y_{k}\alpha\leq\mkern 1.5mu\overline{\mkern-1.5mu\gamma\mkern-1.5mu}\mkern 1.5mu. If ytα<γyt+1αsubscript𝑦𝑡𝛼𝛾subscript𝑦𝑡1𝛼y_{t}\alpha<\gamma\leq y_{t+1}\alpha for some t[k1]𝑡delimited-[]𝑘1t\in[k-1], then (F.2) can be expressed as

𝒞(α,γ)𝒞𝛼𝛾\displaystyle\mathcal{C}(\alpha,\gamma) =tα+m=t+1kγymabsent𝑡𝛼superscriptsubscript𝑚𝑡1𝑘𝛾subscript𝑦𝑚\displaystyle=t\alpha+\sum_{m=t+1}^{k}\frac{\gamma}{y_{m}}
=tα+γ(m=t+1kzm)(nI1)+ϵ(nnI)=tα+stγabsent𝑡𝛼𝛾superscriptsubscript𝑚𝑡1𝑘subscript𝑧𝑚subscript𝑛𝐼1italic-ϵ𝑛subscript𝑛𝐼𝑡𝛼subscript𝑠𝑡𝛾\displaystyle=t\alpha+\frac{\gamma\ (\sum_{m=t+1}^{k}z_{m})}{(n_{I}-1)+\epsilon(n-n_{I})}=t\alpha+s_{t}\gamma

where {st}subscript𝑠𝑡\{s_{t}\} is defined in (23). If 0γy1α0𝛾subscript𝑦1𝛼0\leq\gamma\leq y_{1}\alpha, then

𝒞(α,γ)𝒞𝛼𝛾\displaystyle\mathcal{C}(\alpha,\gamma) =m=1kγym=γ(m=1kzm)(nI1)+ϵ(nnI)=s0γ.absentsuperscriptsubscript𝑚1𝑘𝛾subscript𝑦𝑚𝛾superscriptsubscript𝑚1𝑘subscript𝑧𝑚subscript𝑛𝐼1italic-ϵ𝑛subscript𝑛𝐼subscript𝑠0𝛾\displaystyle=\sum_{m=1}^{k}\frac{\gamma}{y_{m}}=\frac{\gamma\ (\sum_{m=1}^{k}z_{m})}{(n_{I}-1)+\epsilon(n-n_{I})}=s_{0}\gamma.

If ykα<γγ¯subscript𝑦𝑘𝛼𝛾¯𝛾y_{k}\alpha<\gamma\leq\mkern 1.5mu\overline{\mkern-1.5mu\gamma\mkern-1.5mu}\mkern 1.5mu, then 𝒞(α,γ)=m=1kα=kα.𝒞𝛼𝛾superscriptsubscript𝑚1𝑘𝛼𝑘𝛼\mathcal{C}(\alpha,\gamma)=\sum_{m=1}^{k}\alpha=k\alpha. In summary, capacity is

𝒞(α,γ)𝒞𝛼𝛾\displaystyle\mathcal{C}(\alpha,\gamma) ={s0γ,0γy1αtα+stγ,ytα<γyt+1α(t=1,2,,k1)kα,ykα<γγ¯absentcasessubscript𝑠0𝛾0𝛾subscript𝑦1𝛼𝑡𝛼subscript𝑠𝑡𝛾subscript𝑦𝑡𝛼𝛾subscript𝑦𝑡1𝛼otherwise𝑡12𝑘1𝑘𝛼subscript𝑦𝑘𝛼𝛾¯𝛾\displaystyle=\begin{cases}s_{0}\gamma,&0\leq\gamma\leq y_{1}\alpha\\ t\alpha+s_{t}\gamma,&y_{t}\alpha<\gamma\leq y_{t+1}\alpha\\ &\ \ \ \ (t=1,2,\cdots,k-1)\\ k\alpha,&y_{k}\alpha<\gamma\leq\mkern 1.5mu\overline{\mkern-1.5mu\gamma\mkern-1.5mu}\end{cases} (F.3)

which is illustrated in Fig. 29. Since {zt}subscript𝑧𝑡\{z_{t}\} is a decreasing sequence from Remark 1, we have ztzk=(nk)ϵ>0subscript𝑧𝑡subscript𝑧𝑘𝑛𝑘italic-ϵ0z_{t}\geq z_{k}=(n-k)\epsilon>0 for t[k]𝑡delimited-[]𝑘t\in[k]. Thus, {st}t=1ksuperscriptsubscriptsubscript𝑠𝑡𝑡1𝑘\{s_{t}\}_{t=1}^{k} defined in (23) is a monotonically decreasing, non-negative sequence. This implies that the curve in Fig. 29 is a monotonic increasing function of γ𝛾\gamma.

Refer to caption
Figure 29: Capacity as a function of γ𝛾\gamma, for 1nkϵ11𝑛𝑘italic-ϵ1\frac{1}{n-k}\leq\epsilon\leq 1

From Fig. 29, it is shown that 𝒞(α,γ)𝒞𝛼𝛾\mathcal{C}(\alpha,\gamma)\geq\mathcal{M} holds if and only if γγ(α)𝛾superscript𝛾𝛼\gamma\geq\gamma^{*}(\alpha). From (F.3), the threshold value γ(α)superscript𝛾𝛼\gamma^{*}(\alpha) can be expressed as

γ(α)={s0,[0,E1]tαst,(Et,Et+1](t=1,2,,k1),(Ek,).superscript𝛾𝛼casessubscript𝑠00subscript𝐸1𝑡𝛼subscript𝑠𝑡subscript𝐸𝑡subscript𝐸𝑡1otherwise𝑡12𝑘1subscript𝐸𝑘\gamma^{*}(\alpha)=\begin{cases}\frac{\mathcal{M}}{s_{0}},&\mathcal{M}\in[0,E_{1}]\\ \frac{\mathcal{M}-t\alpha}{s_{t}},&\mathcal{M}\in(E_{t},E_{t+1}]\\ &\ \ \ \ (t=1,2,\cdots,k-1)\\ \infty,&\mathcal{M}\in(E_{k},\infty).\end{cases} (F.4)

where

Et=𝒞(α,ytα)=(t1+st1yt)αsubscript𝐸𝑡𝒞𝛼subscript𝑦𝑡𝛼𝑡1subscript𝑠𝑡1subscript𝑦𝑡𝛼E_{t}=\mathcal{C}(\alpha,y_{t}\alpha)=(t-1+s_{t-1}y_{t})\alpha (F.5)

for t[k]𝑡delimited-[]𝑘t\in[k]. The threshold value γ(α)superscript𝛾𝛼\gamma^{*}(\alpha) in (F.4) can be expressed as (20), which completes the proof.

F-A2 Otherwise (if 0ϵ<1nk0italic-ϵ1𝑛𝑘0\leq\epsilon<\frac{1}{n-k})

Using (B.2),

zk=(nk)ϵ<1subscript𝑧𝑘𝑛𝑘italic-ϵ1z_{k}=(n-k)\epsilon<1 (F.6)

holds. Since {zt}subscript𝑧𝑡\{z_{t}\} is a decreasing sequence from Remark 1, there exists τ{0,1,,k1}𝜏01𝑘1\tau\in\{0,1,\cdots,k-1\} such that zτ+1<1zτsubscript𝑧𝜏11subscript𝑧𝜏z_{\tau+1}<1\leq z_{\tau} holds, or equivalently, yταγ¯<yτ+1α.subscript𝑦𝜏𝛼¯𝛾subscript𝑦𝜏1𝛼y_{\tau}\alpha\leq\mkern 1.5mu\overline{\mkern-1.5mu\gamma\mkern-1.5mu}\mkern 1.5mu<y_{\tau+1}\alpha.

Using the analysis similar to the 1nkϵ11𝑛𝑘italic-ϵ1\frac{1}{n-k}\leq\epsilon\leq 1 case, we obtain

𝒞(α,γ)𝒞𝛼𝛾\displaystyle\mathcal{C}(\alpha,\gamma) ={s0γ,0γy1αtα+stγ,ytα<γyt+1α(t=1,2,,τ1)τα+sτγ,yτα<γγ¯absentcasessubscript𝑠0𝛾0𝛾subscript𝑦1𝛼𝑡𝛼subscript𝑠𝑡𝛾subscript𝑦𝑡𝛼𝛾subscript𝑦𝑡1𝛼otherwise𝑡12𝜏1𝜏𝛼subscript𝑠𝜏𝛾subscript𝑦𝜏𝛼𝛾¯𝛾\displaystyle=\begin{cases}s_{0}\gamma,&0\leq\gamma\leq y_{1}\alpha\\ t\alpha+s_{t}\gamma,&y_{t}\alpha<\gamma\leq y_{t+1}\alpha\\ &\ \ \ \ (t=1,2,\cdots,\tau-1)\\ \tau\alpha+s_{\tau}\gamma,&y_{\tau}\alpha<\gamma\leq\mkern 1.5mu\overline{\mkern-1.5mu\gamma\mkern-1.5mu}\end{cases} (F.7)

which is illustrated in Fig. 30.

Refer to caption
Figure 30: Capacity as a function of γ𝛾\gamma, for 0ϵ<1nk0italic-ϵ1𝑛𝑘0\leq\epsilon<\frac{1}{n-k} case

From Fig. 30, it is shown that 𝒞(α,γ)𝒞𝛼𝛾\mathcal{C}(\alpha,\gamma)\geq\mathcal{M} holds if and only if γγ(α)𝛾superscript𝛾𝛼\gamma\geq\gamma^{*}(\alpha). From (F.7), the threshold value γ(α)superscript𝛾𝛼\gamma^{*}(\alpha) can be expressed as

γ(α)={s0,[0,E1]tαst,(Et,Et+1](t=1,2,,τ1)ταsτ,(Eτ,E¯],(E¯,).superscript𝛾𝛼casessubscript𝑠00subscript𝐸1𝑡𝛼subscript𝑠𝑡subscript𝐸𝑡subscript𝐸𝑡1otherwise𝑡12𝜏1𝜏𝛼subscript𝑠𝜏subscript𝐸𝜏¯𝐸¯𝐸\gamma^{*}(\alpha)=\begin{cases}\frac{\mathcal{M}}{s_{0}},&\mathcal{M}\in[0,E_{1}]\\ \frac{\mathcal{M}-t\alpha}{s_{t}},&\mathcal{M}\in(E_{t},E_{t+1}]\\ &\ \ \ \ (t=1,2,\cdots,\tau-1)\\ \frac{\mathcal{M}-\tau\alpha}{s_{\tau}},&\mathcal{M}\in(E_{\tau},\mkern 1.5mu\overline{\mkern-1.5muE\mkern-1.5mu}\mkern 1.5mu]\\ \infty,&\mathcal{M}\in(\mkern 1.5mu\overline{\mkern-1.5muE\mkern-1.5mu}\mkern 1.5mu,\infty).\end{cases} (F.8)

where {Et}subscript𝐸𝑡\{E_{t}\} is defined in (F.5), and

E¯=𝒞(α,γ¯)=τα+sτα{nI1+(nnI)ϵ}.¯𝐸𝒞𝛼¯𝛾𝜏𝛼subscript𝑠𝜏𝛼subscript𝑛𝐼1𝑛subscript𝑛𝐼italic-ϵ\mkern 1.5mu\overline{\mkern-1.5muE\mkern-1.5mu}\mkern 1.5mu=\mathcal{C}(\alpha,\mkern 1.5mu\overline{\mkern-1.5mu\gamma\mkern-1.5mu}\mkern 1.5mu)=\tau\alpha+s_{\tau}\alpha\{n_{I}-1+(n-n_{I})\epsilon\}.

The threshold value γ(α)superscript𝛾𝛼\gamma^{*}(\alpha) in (F.8) can be expressed as (21), which completes the proof.

F-B Proof of Corollary 2

First, we focus on the MSR point illustrated in Fig. 9. From (20), the MSR point for 1nkϵ11𝑛𝑘italic-ϵ1\frac{1}{n-k}\leq\epsilon\leq 1 is

(αmsr(ϵ),γmsr(ϵ))superscriptsubscript𝛼𝑚𝑠𝑟italic-ϵsuperscriptsubscript𝛾𝑚𝑠𝑟italic-ϵ\displaystyle(\alpha_{msr}^{(\epsilon)},\gamma_{msr}^{(\epsilon)}) =(k+skyk,(k1)αmsr(ϵ)sk1)absent𝑘subscript𝑠𝑘subscript𝑦𝑘𝑘1superscriptsubscript𝛼𝑚𝑠𝑟italic-ϵsubscript𝑠𝑘1\displaystyle=(\frac{\mathcal{M}}{k+s_{k}y_{k}},\frac{\mathcal{M}-(k-1)\alpha_{msr}^{(\epsilon)}}{s_{k-1}})
=(k,k1sk1)absent𝑘𝑘1subscript𝑠𝑘1\displaystyle=(\frac{\mathcal{M}}{k},\frac{\mathcal{M}}{k}\frac{1}{s_{k-1}}) (F.9)

where the last equality is from sk=0subscript𝑠𝑘0s_{k}=0 in (23). Moreover, from (21), the MSR point for 0ϵ<1nk0italic-ϵ1𝑛𝑘0\leq\epsilon<\frac{1}{n-k} is

(αmsr(ϵ),γmsr(ϵ))superscriptsubscript𝛼𝑚𝑠𝑟italic-ϵsuperscriptsubscript𝛾𝑚𝑠𝑟italic-ϵ\displaystyle(\alpha_{msr}^{(\epsilon)},\gamma_{msr}^{(\epsilon)}) =(Mτ+i=τ+1kzi,ταmsr(ϵ)sτ)absent𝑀𝜏superscriptsubscript𝑖𝜏1𝑘subscript𝑧𝑖𝜏superscriptsubscript𝛼𝑚𝑠𝑟italic-ϵsubscript𝑠𝜏\displaystyle=(\frac{M}{\tau+\sum_{i=\tau+1}^{k}z_{i}},\frac{\mathcal{M}-\tau\alpha_{msr}^{(\epsilon)}}{s_{\tau}})
=(Mτ+i=τ+1kzi,Mτ+i=τ+1kzii=τ+1kzisτ).absent𝑀𝜏superscriptsubscript𝑖𝜏1𝑘subscript𝑧𝑖𝑀𝜏superscriptsubscript𝑖𝜏1𝑘subscript𝑧𝑖superscriptsubscript𝑖𝜏1𝑘subscript𝑧𝑖subscript𝑠𝜏\displaystyle=(\frac{M}{\tau+\sum_{i=\tau+1}^{k}z_{i}},\frac{M}{\tau+\sum_{i=\tau+1}^{k}z_{i}}\frac{\sum_{i=\tau+1}^{k}z_{i}}{s_{\tau}}). (F.10)

Equations (F.9) and (F.10) proves (28). The expression (29) for the MBR point is directly obtained from Corollary 1 and Fig. 9.

Appendix G Proofs of Propositions

G-A Proof of Proposition 1

As in (A.23), capacity 𝒞𝒞\mathcal{C} for maximum dI,dcsubscript𝑑𝐼subscript𝑑𝑐d_{I},d_{c} setting (dI=nI1,dc=nnIformulae-sequencesubscript𝑑𝐼subscript𝑛𝐼1subscript𝑑𝑐𝑛subscript𝑛𝐼d_{I}=n_{I}-1,d_{c}=n-n_{I}) is expressed as

𝒞=min𝒔S,𝝅Π(𝒔)L(𝒔,𝝅)𝒞subscriptformulae-sequence𝒔𝑆𝝅Π𝒔𝐿𝒔𝝅\mathcal{C}=\displaystyle\min_{\bm{s}\in S,\bm{\pi}\in\Pi(\bm{s})}L(\bm{s},\bm{\pi}) (G.1)

where

L(𝒔,𝝅)𝐿𝒔𝝅\displaystyle L(\bm{s},\bm{\pi}) =i=1kmin{α,ωi(𝝅)},absentsuperscriptsubscript𝑖1𝑘𝛼subscript𝜔𝑖𝝅\displaystyle=\sum_{i=1}^{k}\min\{\alpha,\omega_{i}(\bm{\pi})\},
ωi(𝝅)subscript𝜔𝑖𝝅\displaystyle\omega_{i}(\bm{\pi}) =γei(𝝅)βI(i1ei(𝝅))βc,absent𝛾subscript𝑒𝑖𝝅subscript𝛽𝐼𝑖1subscript𝑒𝑖𝝅subscript𝛽𝑐\displaystyle=\gamma-e_{i}(\bm{\pi})\beta_{I}-(i-1-e_{i}(\bm{\pi}))\beta_{c}, (G.2)
ei(𝝅)subscript𝑒𝑖𝝅\displaystyle e_{i}(\bm{\pi}) =j=1i1𝟙πj=πi,absentsuperscriptsubscript𝑗1𝑖1subscript1subscript𝜋𝑗subscript𝜋𝑖\displaystyle=\sum_{j=1}^{i-1}\mathds{1}_{\pi_{j}=\pi_{i}},

as in (A.20), (A.21) and (A.19).

Consider a general dI,dcsubscript𝑑𝐼subscript𝑑𝑐d_{I},d_{c} setting, where each newcomer node is helped by dIsubscript𝑑𝐼d_{I} nodes in the same cluster, receiving βIsubscript𝛽𝐼\beta_{I} information from each node, and dcsubscript𝑑𝑐d_{c} nodes in other clusters, receiving βcsubscript𝛽𝑐\beta_{c} information from each node. Under this setting, the coefficient of βIsubscript𝛽𝐼\beta_{I} in (G.2) cannot exceed dIsubscript𝑑𝐼d_{I}. Similarly, the coefficient of βcsubscript𝛽𝑐\beta_{c} in (G.2) cannot exceed dcsubscript𝑑𝑐d_{c}. Thus, the capacity for general dI,dcsubscript𝑑𝐼subscript𝑑𝑐d_{I},d_{c} is expressed as

𝒞(dI,dc)=min𝒔S,𝝅Π(𝒔)L(dI,dc,𝒔,𝝅)𝒞subscript𝑑𝐼subscript𝑑𝑐subscriptformulae-sequence𝒔𝑆𝝅Π𝒔𝐿subscript𝑑𝐼subscript𝑑𝑐𝒔𝝅\mathcal{C}(d_{I},d_{c})=\displaystyle\min_{\bm{s}\in S,\bm{\pi}\in\Pi(\bm{s})}\ L(d_{I},d_{c},\bm{s},\bm{\pi}) (G.3)

where

L(dI,dc,𝒔,𝝅)𝐿subscript𝑑𝐼subscript𝑑𝑐𝒔𝝅\displaystyle L(d_{I},d_{c},\bm{s},\bm{\pi}) =i=1kmin{α,ωi(dI,dc,𝒔,𝝅)},absentsuperscriptsubscript𝑖1𝑘𝛼subscript𝜔𝑖subscript𝑑𝐼subscript𝑑𝑐𝒔𝝅\displaystyle=\sum_{i=1}^{k}\min\{\alpha,\omega_{i}(d_{I},d_{c},\bm{s},\bm{\pi})\},
ωi(dI,dc,𝒔,𝝅)subscript𝜔𝑖subscript𝑑𝐼subscript𝑑𝑐𝒔𝝅\displaystyle\omega_{i}(d_{I},d_{c},\bm{s},\bm{\pi}) =γmin{dI,ei(𝝅)}βIabsent𝛾subscript𝑑𝐼subscript𝑒𝑖𝝅subscript𝛽𝐼\displaystyle=\gamma-\min\{d_{I},e_{i}(\bm{\pi})\}\beta_{I}
min{dc,i1ei(𝝅)}βc,subscript𝑑𝑐𝑖1subscript𝑒𝑖𝝅subscript𝛽𝑐\displaystyle\quad\quad-\min\{d_{c},i-1-e_{i}(\bm{\pi})\}\beta_{c}, (G.4)
ei(𝝅)subscript𝑒𝑖𝝅\displaystyle e_{i}(\bm{\pi}) =j=1i1𝟙πj=πi.absentsuperscriptsubscript𝑗1𝑖1subscript1subscript𝜋𝑗subscript𝜋𝑖\displaystyle=\sum_{j=1}^{i-1}\mathds{1}_{\pi_{j}=\pi_{i}}.

Consider arbitrary fixed 𝒔,𝝅𝒔𝝅\bm{s},\bm{\pi} and dcsubscript𝑑𝑐d_{c}. Since γ𝛾\gamma and γc=dcβcsubscript𝛾𝑐subscript𝑑𝑐subscript𝛽𝑐\gamma_{c}=d_{c}\beta_{c} are fixed in the basic setting of Proposition 1, only dIsubscript𝑑𝐼d_{I} and βIsubscript𝛽𝐼\beta_{I} are variables in (G.4), while other parameters are constants. Then, (G.4) can be expressed as

ωi(dI,dc,𝒔,𝝅)subscript𝜔𝑖subscript𝑑𝐼subscript𝑑𝑐𝒔𝝅\displaystyle\omega_{i}(d_{I},d_{c},\bm{s},\bm{\pi}) =C1min{dI,ei(𝝅)}βIabsentsubscript𝐶1subscript𝑑𝐼subscript𝑒𝑖𝝅subscript𝛽𝐼\displaystyle=C_{1}-\min\{d_{I},e_{i}(\bm{\pi})\}\beta_{I}
=C1min{dI,ei(𝝅)}dIC2absentsubscript𝐶1subscript𝑑𝐼subscript𝑒𝑖𝝅subscript𝑑𝐼subscript𝐶2\displaystyle=C_{1}-\frac{\min\{d_{I},e_{i}(\bm{\pi})\}}{d_{I}}C_{2} (G.5)

where C1=γmin{dc,i1ei(𝝅)}βcsubscript𝐶1𝛾subscript𝑑𝑐𝑖1subscript𝑒𝑖𝝅subscript𝛽𝑐C_{1}=\gamma-\min\{d_{c},i-1-e_{i}(\bm{\pi})\}\beta_{c} and C2=γI=γγcsubscript𝐶2subscript𝛾𝐼𝛾subscript𝛾𝑐C_{2}=\gamma_{I}=\gamma-\gamma_{c} are constants. Note that

min{dI,ei(𝝅)}dI={1, if dIei(𝝅),ei(𝝅)dI, otherwise subscript𝑑𝐼subscript𝑒𝑖𝝅subscript𝑑𝐼cases1 if subscript𝑑𝐼subscript𝑒𝑖𝝅subscript𝑒𝑖𝝅subscript𝑑𝐼 otherwise \frac{\min\{d_{I},e_{i}(\bm{\pi})\}}{d_{I}}=\begin{cases}1,&\text{ if }d_{I}\leq e_{i}(\bm{\pi}),\\ \frac{e_{i}(\bm{\pi})}{d_{I}},&\text{ otherwise }\end{cases} (G.6)

is a non-increasing function of dIsubscript𝑑𝐼d_{I}. Thus, ωi(dI,dc,𝒔,𝝅)subscript𝜔𝑖subscript𝑑𝐼subscript𝑑𝑐𝒔𝝅\omega_{i}(d_{I},d_{c},\bm{s},\bm{\pi}) in (G.5) is a non-decreasing function of dIsubscript𝑑𝐼d_{I} for arbitrary fixed 𝒔,𝝅,dc𝒔𝝅subscript𝑑𝑐\bm{s},\bm{\pi},d_{c} and i[k]𝑖delimited-[]𝑘i\in[k]. Since the maximum dIsubscript𝑑𝐼d_{I} value is nI1subscript𝑛𝐼1n_{I}-1, we have

L(dI,dc,𝒔,𝝅)𝐿subscript𝑑𝐼subscript𝑑𝑐𝒔𝝅\displaystyle L(d_{I},d_{c},\bm{s},\bm{\pi}) =i=1kmin{α,ωi(dI,dc,𝒔,𝝅)}absentsuperscriptsubscript𝑖1𝑘𝛼subscript𝜔𝑖subscript𝑑𝐼subscript𝑑𝑐𝒔𝝅\displaystyle=\sum_{i=1}^{k}\min\{\alpha,\omega_{i}(d_{I},d_{c},\bm{s},\bm{\pi})\}
i=1kmin{α,ωi(nI1,dc,𝒔,𝝅)}absentsuperscriptsubscript𝑖1𝑘𝛼subscript𝜔𝑖subscript𝑛𝐼1subscript𝑑𝑐𝒔𝝅\displaystyle\leq\sum_{i=1}^{k}\min\{\alpha,\omega_{i}(n_{I}-1,d_{c},\bm{s},\bm{\pi})\}
=L(nI1,dc,𝒔,𝝅)absent𝐿subscript𝑛𝐼1subscript𝑑𝑐𝒔𝝅\displaystyle=L(n_{I}-1,d_{c},\bm{s},\bm{\pi}) (G.7)

for dI[nI1]subscript𝑑𝐼delimited-[]subscript𝑛𝐼1d_{I}\in[n_{I}-1]. In other words, for all 𝒔,𝝅,dc𝒔𝝅subscript𝑑𝑐\bm{s},\bm{\pi},d_{c}, we have

argmaxdI[nI1]L(dI,dc,𝒔,𝝅)=nI1.subscript𝑑𝐼delimited-[]subscript𝑛𝐼1𝐿subscript𝑑𝐼subscript𝑑𝑐𝒔𝝅subscript𝑛𝐼1\underset{d_{I}\in[n_{I}-1]}{\arg\max}\ L(d_{I},d_{c},\bm{s},\bm{\pi})=n_{I}-1.

Similarly, for all 𝒔,𝝅,dI𝒔𝝅subscript𝑑𝐼\bm{s},\bm{\pi},d_{I},

argmaxdc[nnI]L(dI,dc,𝒔,𝝅)=nnIsubscript𝑑𝑐delimited-[]𝑛subscript𝑛𝐼𝐿subscript𝑑𝐼subscript𝑑𝑐𝒔𝝅𝑛subscript𝑛𝐼\underset{d_{c}\in[n-n_{I}]}{\arg\max}\ L(d_{I},d_{c},\bm{s},\bm{\pi})=n-n_{I}

holds. Therefore, for all 𝒔,𝝅𝒔𝝅\bm{s},\bm{\pi},

argmax[dI,dc]L(dI,dc,𝒔,𝝅)=[nI1,nnI].subscript𝑑𝐼subscript𝑑𝑐𝐿subscript𝑑𝐼subscript𝑑𝑐𝒔𝝅subscript𝑛𝐼1𝑛subscript𝑛𝐼\underset{[d_{I},d_{c}]}{\arg\max}\ L(d_{I},d_{c},\bm{s},\bm{\pi})=[n_{I}-1,n-n_{I}]. (G.8)

Let

[𝒔,𝝅]=argmin𝒔S,𝝅Π(𝒔)L(nI1,nnI,𝒔,𝝅).superscript𝒔superscript𝝅formulae-sequence𝒔𝑆𝝅Π𝒔𝐿subscript𝑛𝐼1𝑛subscript𝑛𝐼𝒔𝝅[\bm{s}^{*},\bm{\pi}^{*}]=\underset{\bm{s}\in S,\bm{\pi}\in\Pi(\bm{s})}{\arg\min}\ \ L(n_{I}-1,n-n_{I},\bm{s},\bm{\pi}). (G.9)

Then, from (G.3), (G.8) and (G.9),

𝒞(dI,dc)𝒞subscript𝑑𝐼subscript𝑑𝑐\displaystyle\mathcal{C}(d_{I},d_{c}) =min𝒔S,𝝅Π(𝒔)L(dI,dc,𝒔,𝝅)absentsubscriptformulae-sequence𝒔𝑆𝝅Π𝒔𝐿subscript𝑑𝐼subscript𝑑𝑐𝒔𝝅\displaystyle=\displaystyle\min_{\bm{s}\in S,\bm{\pi}\in\Pi(\bm{s})}\ L(d_{I},d_{c},\bm{s},\bm{\pi})
L(dI,dc,𝒔,𝝅)L(nI1,nnI,𝒔,𝝅)absent𝐿subscript𝑑𝐼subscript𝑑𝑐superscript𝒔superscript𝝅𝐿subscript𝑛𝐼1𝑛subscript𝑛𝐼superscript𝒔superscript𝝅\displaystyle\leq L(d_{I},d_{c},\bm{s}^{*},\bm{\pi}^{*})\leq L(n_{I}-1,n-n_{I},\bm{s}^{*},\bm{\pi}^{*})
=min𝒔S,𝝅Π(𝒔)L(nI1,nnI,𝒔,𝝅)absentsubscriptformulae-sequence𝒔𝑆𝝅Π𝒔𝐿subscript𝑛𝐼1𝑛subscript𝑛𝐼𝒔𝝅\displaystyle=\displaystyle\min_{\bm{s}\in S,\bm{\pi}\in\Pi(\bm{s})}\ L(n_{I}-1,n-n_{I},\bm{s},\bm{\pi})
=𝒞(nI1,nnI)absent𝒞subscript𝑛𝐼1𝑛subscript𝑛𝐼\displaystyle=\mathcal{C}(n_{I}-1,n-n_{I}) (G.10)

for all dI[nI1]subscript𝑑𝐼delimited-[]subscript𝑛𝐼1d_{I}\in[n_{I}-1] and dc[nnI]subscript𝑑𝑐delimited-[]𝑛subscript𝑛𝐼d_{c}\in[n-n_{I}]. Therefore, choosing dI=nI1subscript𝑑𝐼subscript𝑛𝐼1d_{I}=n_{I}-1 and dc=nnIsubscript𝑑𝑐𝑛subscript𝑛𝐼d_{c}=n-n_{I} maximizes storage capacity when the available resources, γ𝛾\gamma and γcsubscript𝛾𝑐\gamma_{c}, are given.

G-B Proof of Proposition 2

First, we prove (8). Recall ρisubscript𝜌𝑖\rho_{i} and gmsubscript𝑔𝑚g_{m} defined in (6) and (7). Consider the support set S𝑆S, which is defined as

S={i[nI]:gi1}.𝑆conditional-set𝑖delimited-[]subscript𝑛𝐼subscript𝑔𝑖1S=\{i\in[n_{I}]:g_{i}\geq 1\}. (G.11)

Then, we have

ρi=nIisubscript𝜌𝑖subscript𝑛𝐼𝑖\displaystyle\rho_{i}=n_{I}-i nI1,absentsubscript𝑛𝐼1\displaystyle\leq n_{I}-1, (G.12)
m=1i1gmsuperscriptsubscript𝑚1𝑖1subscript𝑔𝑚\displaystyle\sum_{m=1}^{i-1}g_{m} i1absent𝑖1\displaystyle\geq i-1 (G.13)

for every iS𝑖𝑆i\in S. Therefore, by combining (G.13) and (6),

nρijm=1i1gm𝑛subscript𝜌𝑖𝑗superscriptsubscript𝑚1𝑖1subscript𝑔𝑚\displaystyle n-\rho_{i}-j-\sum_{m=1}^{i-1}g_{m} n(i1)jρiabsent𝑛𝑖1𝑗subscript𝜌𝑖\displaystyle\leq n-(i-1)-j-\rho_{i}
=nnI(j1)nnIabsent𝑛subscript𝑛𝐼𝑗1𝑛subscript𝑛𝐼\displaystyle=n-n_{I}-(j-1)\leq n-n_{I} (G.14)

holds for every iS,j[gi]formulae-sequence𝑖𝑆𝑗delimited-[]subscript𝑔𝑖i\in S,j\in[g_{i}]. Combining (2), (G.12) and (G-B) results in

ρiβI+(nρijm=1i1gm)βc(nI1)βI+(nnI)βc=γsubscript𝜌𝑖subscript𝛽𝐼𝑛subscript𝜌𝑖𝑗superscriptsubscript𝑚1𝑖1subscript𝑔𝑚subscript𝛽𝑐subscript𝑛𝐼1subscript𝛽𝐼𝑛subscript𝑛𝐼subscript𝛽𝑐𝛾\rho_{i}\beta_{I}+(n-\rho_{i}-j-\sum_{m=1}^{i-1}g_{m})\beta_{c}\leq(n_{I}-1)\beta_{I}+(n-n_{I})\beta_{c}=\gamma (G.15)

for arbitrary iS,j[gi]formulae-sequence𝑖𝑆𝑗delimited-[]subscript𝑔𝑖i\in S,j\in[g_{i}]. Since [gi]=delimited-[]subscript𝑔𝑖[g_{i}]=\emptyset holds for i[nI]S𝑖delimited-[]subscript𝑛𝐼𝑆i\in[n_{I}]\setminus S, we conclude that (G.15) holds for (i,j)𝑖𝑗(i,j) with i[nI]𝑖delimited-[]subscript𝑛𝐼i\in[n_{I}], j[gi]𝑗delimited-[]subscript𝑔𝑖j\in[g_{i}].

Second, we prove (9). Using q𝑞q and r𝑟r in (C.9) and (C.10),

gi={q+1,irq,otherwisesubscript𝑔𝑖cases𝑞1𝑖𝑟𝑞otherwiseg_{i}=\begin{cases}q+1,&i\leq r\\ q,&\text{otherwise}\end{cases} (G.16)

Therefore, i=1nIgi=(q+1)r+q(nIr)=r+qnI=k,superscriptsubscript𝑖1subscript𝑛𝐼subscript𝑔𝑖𝑞1𝑟𝑞subscript𝑛𝐼𝑟𝑟𝑞subscript𝑛𝐼𝑘\sum_{i=1}^{n_{I}}g_{i}=(q+1)r+q(n_{I}-r)=r+qn_{I}=k, where the last equality is from (C.11).

Appendix H Proofs of Lemmas

H-A Proof of Lemma 1

Using (2), C¯¯𝐶\underline{C} in (14) can be expressed as

C¯¯𝐶\displaystyle\underline{C} =k2((nI1)βI+(nnI)(1+nkn(11/L))βc)absent𝑘2subscript𝑛𝐼1subscript𝛽𝐼𝑛subscript𝑛𝐼1𝑛𝑘𝑛11𝐿subscript𝛽𝑐\displaystyle=\frac{k}{2}\left((n_{I}-1)\beta_{I}+(n-n_{I})(1+\frac{n-k}{n(1-1/L)})\beta_{c}\right)
=k2((nI1)βI+(nnI)(1+nknnI)βc)absent𝑘2subscript𝑛𝐼1subscript𝛽𝐼𝑛subscript𝑛𝐼1𝑛𝑘𝑛subscript𝑛𝐼subscript𝛽𝑐\displaystyle=\frac{k}{2}\left((n_{I}-1)\beta_{I}+(n-n_{I})(1+\frac{n-k}{n-n_{I}})\beta_{c}\right)
=k2{(nI1)βI+(2nnIk)βc}.absent𝑘2subscript𝑛𝐼1subscript𝛽𝐼2𝑛subscript𝑛𝐼𝑘subscript𝛽𝑐\displaystyle=\frac{k}{2}\left\{(n_{I}-1)\beta_{I}+(2n-n_{I}-k)\beta_{c}\right\}. (H.1)

According to (A.20) and (A.68) in Appendix B, the capacity can be expressed as

𝒞=L(𝒔h,𝝅v)=i=1kmin{α,ωi(𝝅v(𝒔h))}.𝒞𝐿subscript𝒔subscript𝝅𝑣superscriptsubscript𝑖1𝑘𝛼subscript𝜔𝑖subscript𝝅𝑣subscript𝒔\mathcal{C}=L(\bm{s}_{h},\bm{\pi}_{v})=\sum_{i=1}^{k}\min\{\alpha,\omega_{i}(\bm{\pi}_{v}(\bm{s}_{h}))\}. (H.2)

From (A.21), we have

ωi(𝝅v(𝒔h))γsubscript𝜔𝑖subscript𝝅𝑣subscript𝒔𝛾\omega_{i}(\bm{\pi}_{v}(\bm{s}_{h}))\leq\gamma (H.3)

for i[k]𝑖delimited-[]𝑘i\in[k]. Therefore, when α=γ𝛼𝛾\alpha=\gamma, the capacity expression in (H.2) reduces to

𝒞=i=1kωi(𝝅v(𝒔h)).𝒞superscriptsubscript𝑖1𝑘subscript𝜔𝑖subscript𝝅𝑣subscript𝒔\mathcal{C}=\sum_{i=1}^{k}\omega_{i}(\bm{\pi}_{v}(\bm{s}_{h})). (H.4)

Recall (A.64):

ωi(𝝅v(𝒔h))=(nIhi)βI+(ninI+hi)βc.subscript𝜔𝑖subscript𝝅𝑣subscript𝒔subscript𝑛𝐼subscript𝑖subscript𝛽𝐼𝑛𝑖subscript𝑛𝐼subscript𝑖subscript𝛽𝑐\omega_{i}(\bm{\pi}_{v}(\bm{s}_{h}))=(n_{I}-h_{i})\beta_{I}+(n-i-n_{I}+h_{i})\beta_{c}. (H.5)

From the expression of (hi)i=1ksuperscriptsubscriptsubscript𝑖𝑖1𝑘(h_{i})_{i=1}^{k} in (A.66), we have

i=1k(nIhi)=s=1nIgs(nIs)superscriptsubscript𝑖1𝑘subscript𝑛𝐼subscript𝑖superscriptsubscript𝑠1subscript𝑛𝐼subscript𝑔𝑠subscript𝑛𝐼𝑠\sum_{i=1}^{k}(n_{I}-h_{i})=\sum_{s=1}^{n_{I}}g_{s}(n_{I}-s) (H.6)

since m=1kgm=ksuperscriptsubscript𝑚1𝑘subscript𝑔𝑚𝑘\sum_{m=1}^{k}g_{m}=k from (9),

Using (H.5) and (H.6), the capacity expression in (H.4) is expressed as

𝒞𝒞\displaystyle\mathcal{C} =(i=1k(nIhi))βI+(i=1k(ni(nIhi)))βcabsentsuperscriptsubscript𝑖1𝑘subscript𝑛𝐼subscript𝑖subscript𝛽𝐼superscriptsubscript𝑖1𝑘𝑛𝑖subscript𝑛𝐼subscript𝑖subscript𝛽𝑐\displaystyle=\left(\sum_{i=1}^{k}(n_{I}-h_{i})\right)\beta_{I}+\left(\sum_{i=1}^{k}(n-i-(n_{I}-h_{i}))\right)\beta_{c}
=A0βI+B0βcabsentsubscript𝐴0subscript𝛽𝐼subscript𝐵0subscript𝛽𝑐\displaystyle=A_{0}\beta_{I}+B_{0}\beta_{c} (H.7)

where A0=s=1nIgs(nIs)subscript𝐴0superscriptsubscript𝑠1subscript𝑛𝐼subscript𝑔𝑠subscript𝑛𝐼𝑠A_{0}=\sum_{s=1}^{n_{I}}g_{s}(n_{I}-s) and

B0subscript𝐵0\displaystyle B_{0} =i=1k(ni)A0.absentsuperscriptsubscript𝑖1𝑘𝑛𝑖subscript𝐴0\displaystyle=\sum_{i=1}^{k}(n-i)-A_{0}. (H.8)

Similarly, C¯¯𝐶\underline{C} in (H.1) can be expressed as

C¯=A0βI+B0βc¯𝐶superscriptsubscript𝐴0subscript𝛽𝐼superscriptsubscript𝐵0subscript𝛽𝑐\underline{C}=A_{0}^{\prime}\beta_{I}+B_{0}^{\prime}\beta_{c} (H.9)

where A0=(nI1)k2superscriptsubscript𝐴0subscript𝑛𝐼1𝑘2A_{0}^{\prime}=(n_{I}-1)\frac{k}{2} and

B0superscriptsubscript𝐵0\displaystyle B_{0}^{\prime} =(2nnIk)k2=i=1k(ni)A0.absent2𝑛subscript𝑛𝐼𝑘𝑘2superscriptsubscript𝑖1𝑘𝑛𝑖superscriptsubscript𝐴0\displaystyle=(2n-n_{I}-k)\frac{k}{2}=\sum_{i=1}^{k}(n-i)-A_{0}^{\prime}. (H.10)

First, we show that 𝒞C¯𝒞¯𝐶\mathcal{C}\geq\underline{C} holds. From (H-A), (H.8), (H.9) and (H.10),

𝒞𝒞\displaystyle\mathcal{C} C¯=(A0A0)βI+(B0B0)βc¯𝐶subscript𝐴0superscriptsubscript𝐴0subscript𝛽𝐼subscript𝐵0superscriptsubscript𝐵0subscript𝛽𝑐\displaystyle-\underline{C}=(A_{0}-A_{0}^{\prime})\beta_{I}+(B_{0}-B_{0}^{\prime})\beta_{c}
=(A0A0)βI(A0A0)βc=(A0A0)(βIβc).absentsubscript𝐴0superscriptsubscript𝐴0subscript𝛽𝐼subscript𝐴0superscriptsubscript𝐴0subscript𝛽𝑐subscript𝐴0superscriptsubscript𝐴0subscript𝛽𝐼subscript𝛽𝑐\displaystyle=(A_{0}-A_{0}^{\prime})\beta_{I}-(A_{0}-A_{0}^{\prime})\beta_{c}=(A_{0}-A_{0}^{\prime})(\beta_{I}-\beta_{c}). (H.11)

Since we consider the βIβcsubscript𝛽𝐼subscript𝛽𝑐\beta_{I}\geq\beta_{c} case, all we need to prove is

A0A00.subscript𝐴0superscriptsubscript𝐴00A_{0}-A_{0}^{\prime}\geq 0.

Using (gi)i=1ksuperscriptsubscriptsubscript𝑔𝑖𝑖1𝑘(g_{i})_{i=1}^{k} expression in (G.16), A0subscript𝐴0A_{0} can be rewritten as

A0subscript𝐴0\displaystyle A_{0} =s=1nIgs(nIs)=qs=1nI(nIs)+s=1r(nIs)absentsuperscriptsubscript𝑠1subscript𝑛𝐼subscript𝑔𝑠subscript𝑛𝐼𝑠𝑞superscriptsubscript𝑠1subscript𝑛𝐼subscript𝑛𝐼𝑠superscriptsubscript𝑠1𝑟subscript𝑛𝐼𝑠\displaystyle=\sum_{s=1}^{n_{I}}g_{s}(n_{I}-s)=q\sum_{s=1}^{n_{I}}(n_{I}-s)+\sum_{s=1}^{r}(n_{I}-s)
=qnI(nI1)2+s=1r(nIs)absent𝑞subscript𝑛𝐼subscript𝑛𝐼12superscriptsubscript𝑠1𝑟subscript𝑛𝐼𝑠\displaystyle=q\frac{n_{I}(n_{I}-1)}{2}+\sum_{s=1}^{r}(n_{I}-s)
=qnI(nI1)2+r((nI1)+(nIr))2absent𝑞subscript𝑛𝐼subscript𝑛𝐼12𝑟subscript𝑛𝐼1subscript𝑛𝐼𝑟2\displaystyle=q\frac{n_{I}(n_{I}-1)}{2}+\frac{r((n_{I}-1)+(n_{I}-r))}{2}
=(qnI+r)nI12+r(nIr)2absent𝑞subscript𝑛𝐼𝑟subscript𝑛𝐼12𝑟subscript𝑛𝐼𝑟2\displaystyle=\left(qn_{I}+r\right)\frac{n_{I}-1}{2}+\frac{r(n_{I}-r)}{2}
=k(nI1)2+r(nIr)2=A0+r(nIr)2.absent𝑘subscript𝑛𝐼12𝑟subscript𝑛𝐼𝑟2superscriptsubscript𝐴0𝑟subscript𝑛𝐼𝑟2\displaystyle=\frac{k(n_{I}-1)}{2}+\frac{r(n_{I}-r)}{2}=A_{0}^{\prime}+\frac{r(n_{I}-r)}{2}.

Since 0r<nI0𝑟subscript𝑛𝐼0\leq r<n_{I}, we have

A0A0=r(nIr)20.subscript𝐴0superscriptsubscript𝐴0𝑟subscript𝑛𝐼𝑟20A_{0}-A_{0}^{\prime}=\frac{r(n_{I}-r)}{2}\geq 0. (H.12)

Therefore, 𝒞C¯𝒞¯𝐶\mathcal{C}\geq\underline{C} holds.

Second, we prove 𝒞C¯+nI2(βIβc)/8𝒞¯𝐶superscriptsubscript𝑛𝐼2subscript𝛽𝐼subscript𝛽𝑐8\mathcal{C}\leq\underline{C}+n_{I}^{2}(\beta_{I}-\beta_{c})/8. Note that A0A0subscript𝐴0superscriptsubscript𝐴0A_{0}-A_{0}^{\prime} in (H.12) is maximized when r=nI/2𝑟subscript𝑛𝐼2r=\lfloor n_{I}/2\rfloor holds. Thus, A0A0nI/2(nInI/2)2nI2/8.subscript𝐴0superscriptsubscript𝐴0subscript𝑛𝐼2subscript𝑛𝐼subscript𝑛𝐼22superscriptsubscript𝑛𝐼28A_{0}-A_{0}^{\prime}\leq\frac{\lfloor n_{I}/2\rfloor(n_{I}-\lfloor n_{I}/2\rfloor)}{2}\leq n_{I}^{2}/8. Combining with (H-A),

𝒞C¯𝒞¯𝐶\displaystyle\mathcal{C}-\underline{C} =(A0A0)(βIβc)nI2(βIβc)/8.absentsubscript𝐴0superscriptsubscript𝐴0subscript𝛽𝐼subscript𝛽𝑐superscriptsubscript𝑛𝐼2subscript𝛽𝐼subscript𝛽𝑐8\displaystyle=(A_{0}-A_{0}^{\prime})(\beta_{I}-\beta_{c})\leq n_{I}^{2}(\beta_{I}-\beta_{c})/8.

H-B Proof of Lemma 2

Recall that ξ=γc/γ,γ𝜉subscript𝛾𝑐𝛾𝛾\xi=\gamma_{c}/\gamma,\gamma and R=k/n𝑅𝑘𝑛R=k/n value are all fixed. The expression for C¯¯𝐶\underline{C} in (14) can be expressed as

C¯¯𝐶\displaystyle\underline{C} =k2(γ+nkn(11L)γc)=nR2(γ+1R11Lγc)absent𝑘2𝛾𝑛𝑘𝑛11𝐿subscript𝛾𝑐𝑛𝑅2𝛾1𝑅11𝐿subscript𝛾𝑐\displaystyle=\frac{k}{2}(\gamma+\frac{n-k}{n(1-\frac{1}{L})}\gamma_{c})=\frac{nR}{2}(\gamma+\frac{1-R}{1-\frac{1}{L}}\gamma_{c})
=γnR2(1+1R(11L)ξ)absent𝛾𝑛𝑅211𝑅11𝐿𝜉\displaystyle=\gamma\frac{nR}{2}(1+\frac{1-R}{(1-\frac{1}{L})}\xi) (H.13)

Note that (H.13) is a monotonic decreasing function of L𝐿L. Moreover, we consider the L2𝐿2L\geq 2 case, as mentioned in (4). Thus, C¯¯𝐶\underline{C} is upper/lower bounded by expressions for L=2𝐿2L=2 and L=𝐿L=\infty, respectively:

γnR2(1+(1R)ξ)<C¯γnR2(1+2(1R)ξ).𝛾𝑛𝑅211𝑅𝜉¯𝐶𝛾𝑛𝑅2121𝑅𝜉\gamma\frac{nR}{2}(1+(1-R)\xi)<\underline{C}\leq\gamma\frac{nR}{2}(1+2(1-R)\xi). (H.14)

Therefore, C¯=Θ(n)¯𝐶Θ𝑛\underline{C}=\Theta(n) holds. Moreover, the expression for δ𝛿\delta in (17) is

δ𝛿\displaystyle\delta =nI2(βIβc)8=nI28(γInI1γcnnI)absentsuperscriptsubscript𝑛𝐼2subscript𝛽𝐼subscript𝛽𝑐8superscriptsubscript𝑛𝐼28subscript𝛾𝐼subscript𝑛𝐼1subscript𝛾𝑐𝑛subscript𝑛𝐼\displaystyle=\frac{n_{I}^{2}(\beta_{I}-\beta_{c})}{8}=\frac{n_{I}^{2}}{8}(\frac{\gamma_{I}}{n_{I}-1}-\frac{\gamma_{c}}{n-n_{I}})
=nI28(γ(1ξ)nI1γξnnI)absentsuperscriptsubscript𝑛𝐼28𝛾1𝜉subscript𝑛𝐼1𝛾𝜉𝑛subscript𝑛𝐼\displaystyle=\frac{n_{I}^{2}}{8}(\frac{\gamma(1-\xi)}{n_{I}-1}-\frac{\gamma\xi}{n-n_{I}})
=nI2γ8(1ξ)(nnI)ξ(nI1)(nI1)(nnI).absentsuperscriptsubscript𝑛𝐼2𝛾81𝜉𝑛subscript𝑛𝐼𝜉subscript𝑛𝐼1subscript𝑛𝐼1𝑛subscript𝑛𝐼\displaystyle=\frac{n_{I}^{2}\gamma}{8}\frac{(1-\xi)(n-n_{I})-\xi(n_{I}-1)}{(n_{I}-1)(n-n_{I})}. (H.15)

Putting n=nIL𝑛subscript𝑛𝐼𝐿n=n_{I}L into (H.15), we get

δ𝛿\displaystyle\delta =nIγ8(1ξ)nI(L1)ξ(nI1)(nI1)(L1)=O(nI).absentsubscript𝑛𝐼𝛾81𝜉subscript𝑛𝐼𝐿1𝜉subscript𝑛𝐼1subscript𝑛𝐼1𝐿1𝑂subscript𝑛𝐼\displaystyle=\frac{n_{I}\gamma}{8}\frac{(1-\xi)n_{I}(L-1)-\xi(n_{I}-1)}{(n_{I}-1)(L-1)}=O(n_{I}).

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Jy-yong Sohn (S’15) received the B.S. and M.S. degrees in electrical engineering from the Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea, in 2014 and 2016, respectively. He is currently pursuing the Ph.D. degree in KAIST. His research interests include coding for distributed storage and distributed computing, massive MIMO effects on wireless multi cellular system and 5G Communications. He is a recipient of the IEEE international conference on communications (ICC) best paper award in 2017.
Beongjun Choi (S’17) received the B.S. and M.S. degrees in mathematics and electrical engineering from the Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea, in 2014 and 2017. He is currently pursuing the electrical engineering Ph.D degree in KAIST. His research interests include error-correcting codes, distributed storage system and information theory. He is a co-recipient of the IEEE international conference on communications (ICC) best paper award in 2017.
Sung Whan Yoon (M’17) received the M.S. and Ph.D. degrees in electrical engineering from the Korea Advanced Institute of Science and Technology (KAIST), Daejeon, South Korea, in 2013 and 2017 respectively. He is currently a postdoctoral researcher in KAIST from 2017. His research interests are in the area of coding theory, distributed system and artificial intelligence, with focusing on polar codes, distributed storage system and meta-learning algorithm of neural network. Especially for the area of artificial intelligence, his primary interests include information theoretic analysis and algorithmic development of meta-learning. He was a co-recipient of the IEEE International Conference on Communications best Paper Award in 2017.
Jaekyun Moon (F’05) received the Ph.D degree in electrical and computer engineering at Carnegie Mellon University, Pittsburgh, Pa, USA. He is currently a Professor of electrical engineering at KAIST. From 1990 through early 2009, he was with the faculty of the Department of Electrical and Computer Engineering at the University of Minnesota, Twin Cities. He consulted as Chief Scientist for DSPG, Inc. from 2004 to 2007. He also worked as Chief Technology Officer at Link-A-Media Devices Corporation. His research interests are in the area of channel characterization, signal processing and coding for data storage and digital communication. Prof. Moon received the McKnight Land-Grant Professorship from the University of Minnesota. He received the IBM Faculty Development Awards as well as the IBM Partnership Awards. He was awarded the National Storage Industry Consortium (NSIC) Technical Achievement Award for the invention of the maximum transition run (MTR) code, a widely used error-control/modulation code in commercial storage systems. He served as Program Chair for the 1997 IEEE Magnetic Recording Conference. He is also Past Chair of the Signal Processing for Storage Technical Committee of the IEEE Communications Society. He served as a guest editor for the 2001 IEEE JSAC issue on Signal Processing for High Density Recording. He also served as an Editor for IEEE TRANSACTIONS ON MAGNETICS in the area of signal processing and coding for 2001-2006. He is an IEEE Fellow.